Just to expand on this and add some book quotes, the text makes it very clear that the rejection of their wolfish nature and instincts is part of what leads to both Jon and Robbâs deaths. For Robb, since the beginning of ASOS, he starts ignoring the warnings Grey Wind gives, against Catelynâs advice:
âI am not a wolf, no matter what they call me.â Robb sounded cross. âGrey Wind killed a man at the Crag, another at Ashemark, and six or seven at Oxcross. If you had seenââ
âI saw Branâs wolf tear out a manâs throat at Winterfell,â she said sharply, âand loved him for it.â
âThatâs different. The man at the Crag was a knight Jeyne had known all her life. You canât blame her for being afraid. Grey Wind doesnât like her uncle either. He bares his teeth every time Ser Rolph comes near him.â
A chill went through her. âSend Ser Rolph away. At once.â
âWhere? Back to the Crag, so the Lannisters can mount his head on a spike? Jeyne loves him. Heâs her uncle, and a fair knight besides. I need more men like Rolph Spicer, not fewer. I am not going to banish him just because my wolf doesnât seem to like the way he smells.â
âRobb.â She stopped and held his arm. âI told you once to keep Theon Greyjoy close, and you did not listen. Listen now. Send this man away. I am not saying you must banish him. Find some task that requires a man of courage, some honorable duty, what it is matters not ⌠but do not keep him near you.â
He frowned. âShould I have Grey Wind sniff all my knights? There might be others whose smell he mislikes.â
âAny man Grey Wind mislikes is a man I do not want close to you. These wolves are more than wolves, Robb. You must know that. I think perhaps the gods sent them to us. Your fatherâs gods, the old gods of the north. Five wolf pups, Robb, five for five Stark children.â
âSix,â said Robb. âThere was a wolf for Jon as well. I found them, remember? I know how many there were and where they came from. I used to think the same as you, that the wolves were our guardians, our protectors, until âŚâ
âUntil?â she prompted.
Robbâs mouth tightened. â⌠.until they told me that Theon had murdered Bran and Rickon. Small good their wolves did them. I am no longer a boy, Mother. Iâm a king, and I can protect myself.â He sighed. âI will find some duty for Ser Rolph, some pretext to send him away. Not because of his smell, but to ease your mind. You have suffered enough.â - Catelyn II ASOS
Later we learn that Ser Rolph was part of the ploy with the Freys, and that he gains a lordship for it:
Ser Kevan presented another sheaf of parchments to the king. Tommen dipped and signed. âThis is a decree of legitimacy for a natural son of Lord Roose Bolton of the Dreadfort. And this names Lord Bolton your Warden of the North.â Tommen dipped, signed, dipped, signed. âThis grants Ser Rolph Spicer title to the castle Castamere and raises him to the rank of lord.â Tommen scrawled his name. - Jaime IX ASOS
Grey Wind also doesnât want to go into the Twins:
There was more trouble at the gatehouse. Grey Wind balked in the middle of the drawbridge, shook the rain off, and howled at the portcullis. Robb whistled impatiently. âGrey Wind. What is it? Grey Wind, with me.â But the direwolf only bared his teeth. He does not like this place, Catelyn thought. Robb had to squat and speak softly to the wolf before he would consent to pass beneath the portcullis. - Catelyn VI ASOS
Walder Frey refuses to allow Grey Wind into the castle, meaning that Grey Wind is away from Robb when the Red Wedding happens:
The sight of the dogs made Catelyn wish once more for Grey Wind, but Robbâs direwolf was nowhere to be seen. Lord Walder had refused to allow him in the hall. âYour wild beast has a taste for human flesh, I hear, heh,â the old man had said. âRips out throats, yes. Iâll have no such creature at my Roslinâs feast, amongst women and little ones, all my sweet innocents.â
âGrey Wind is no danger to them, my lord,â Robb protested. âNot so long as I am there.â
âYou were there at my gates, were you not? When the wolf attacked the grandsons I sent to greet you? I heard all about that, donât think I didnât, heh.â
âNo harm was doneââ
âNo harm, the king says? No harm? Petyr fell from his horse, fell. I lost a wife the same way, falling.â His mouth worked in and out. âOr was she just some strumpet? Bastard Walderâs mother, yes, now I recall. She fell off her horse and cracked her head. What would Your Grace do if Petyr had broken his neck, heh? Give me another apology in place of a grandson? No, no, no. Might be youâre king, I wonât say youâre not, the King in the North, heh, but under my roof, my rule. Have your wolf or have your wedding, sire. Youâll not have both.â - Catelyn VII ASOS
Then the tabletop that the Smalljon had flung over Robb shifted, and her son struggled to his knees. He had an arrow in his side, a second in his leg, a third through his chest. Lord Walder raised a hand, and the music stopped, all but one drum. Catelyn heard the crash of distant battle, and closer the wild howling of a wolf. Grey Wind, she remembered too late. - Catelyn VII ASOS
The same happens with Jon. Right at the beginning of ADWD, he is warned by Melisandre to keep his wolf close:
âDo not be so certain.â The ruby at Melisandreâs throat gleamed red. âIt is not the foes who curse you to your face that you must fear, but those who smile when you are looking and sharpen their knives when you turn your back. You would do well to keep your wolf close beside you. Ice, I see, and daggers in the dark. Blood frozen red and hard, and naked steel. It was very cold.â - Jon I ADWD
Then, we see Jon rejecting his wolf nature and claiming to not be a wolf, just like Robb claimed he was not a wolf, and he does this again against Melisandreâs advice:
He was walking beneath the shell of the Lord Commanderâs Tower, past the spot where Ygritte had died in his arms, when Ghost appeared beside him, his warm breath steaming in the cold. In the moonlight, his red eyes glowed like pools of fire. The taste of hot blood filled Jonâs mouth, and he knew that Ghost had killed that night. No, he thought. I am a man, not a wolf. He rubbed his mouth with the back of a gloved hand and spat. - Jon III ADWD
âYou think so?â She knelt and scratched Ghost behind his ear. âYour Wall is a queer place, but there is power here, if you will use it. Power in you, and in this beast. You resist it, and that is your mistake. Embrace it. Use it.â
I am not a wolf, he thought. âAnd how would I do that?â - Jon VI ADWD
Then, right before he is killed, he locks Ghost away, just like Robb was away from Grey Wind when he died:
Horse and Rory had replaced Fulk and Mully at the armory door with the change of watch. âWith me,â Jon told them, when the time came. Ghost would have followed as well, but as the wolf came padding after them, Jon grabbed him by the scruff of his neck and wrestled him back inside. Borroq might be amongst those gathering at the Shieldhall. The last thing he needed just now was his wolf savaging the skinchangerâs boar. - Jon XIII ADWD
So both Jon and Robb die because they reject their wolfish natures and are away from their wolves. And they realize this in the end, as their last words are the names of their direwolves:
âJeyne?â Robb grabbed the edge of the table and forced himself to stand. âMother,â he said, âGrey Wind âŚâ - Catelyn VII ASOS
Jon fell to his knees. He found the daggerâs hilt and wrenched it free. In the cold night air the wound was smoking. âGhost,â he whispered. Pain washed over him. Stick them with the pointy end. When the third dagger took him between the shoulder blades, he gave a grunt and fell face-first into the snow. He never felt the fourth knife. Only the cold ⌠- Jon XIII ADWD
For Robb, his direwolf wasnât able to save him, because they Frey men killed him. For Jon, on the other hand, his reconnection with his wolf will be his salvation.
With Arya, we see this play out with the way she feels like different creatures. First she is a a sheep, then a scared mouse. Then she is a ghost. And she finally frees herself when she reclaims her wolf nature:
On the road Arya had felt like a sheep, but Harrenhal turned her into a mouse. She was grey as a mouse in her scratchy wool shift, and like a mouse she kept to the crannies and crevices and dark holes of the castle, scurrying out of the way of the mighty. - Arya VII ACOK
Arya chewed her lip and tried to think when her courage had come back. Jaqen made me brave again. He made me a ghost instead of a mouse. - Arya IX ACOK
âYou are Arya of Winterfell, daughter of the north. You told me you could be strong. You have the wolf blood in you.â
âThe wolf blood.â Arya remembered now. âIâll be as strong as Robb. I said I would.â She took a deep breath, then lifted the broomstick in both hands and brought it down across her knee. It broke with a loud crack, and she threw the pieces aside. I am a direwolf, and done with wooden teeth. - Arya X ACOK
The text makes it very clear with these examples that rejecting oneâs own nature leads to doom, and accepting it leads to salvation. And the same can be seen with Dany. Locking away her dragons and making peace with the slavers represents Dany being imprisoned and enslaved, compromising her values, which goes against her nature:
Four hours later, they emerged again as man and wife, bound together wrist and ankle with chains of yellow gold. - Daenerys VII ADWD
It was time, though. A girl might spend her life at play, but she was a woman grown, a queen, a wife, a mother to thousands. Her children had need of her. Drogon had bent before the whip, and so must she. She had to don her crown again and return to her ebon bench and the arms of her noble husband. - Daenerys X ADWD
Meanwhile, the dragons represent freedom, which is Danyâs true nature:
âThere is a reason. A dragon is no slave.â And Dany swept the lash down as hard as she could across the slaverâs face. Kraznys screamed and staggered back, the blood running red down his cheeks into his perfumed beard. The harpyâs fingers had torn his features half to pieces with one slash, but she did not pause to contemplate the ruin. âDrogon,â she sang out loudly, sweetly, all her fear forgotten. âDracarys.â - Daenerys III ASOS
âUnsullied!â Dany galloped before them, her silver-gold braid flying behind her, her bell chiming with every stride. âSlay the Good Masters, slay the soldiers, slay every man who wears a tokar or holds a whip, but harm no child under twelve, and strike the chains off every slave you see.â She raised the harpyâs fingers in the air ⌠and then she flung the scourge aside. âFreedom!â she sang out. âDracarys! Dracarys!â
âDracarys!â they shouted back, the sweetest word sheâd ever heard. âDracarys! Dracarys!â And all around them slavers ran and sobbed and begged and died, and the dusty air was filled with spears and fire. - Daenerys III ASOS
This dichotomy is made even more clear when we see that going back to her husband in Meereen (back to the peace with the slavers) and turning away from Drogon is described as âclosing her ears to the song of flight and freedomâ:
No, Dany told herself. If I look back I am lost. She might live for years amongst the sunbaked rocks of Dragonstone, riding Drogon by day and gnawing at his leavings every evenfall as the great grass sea turned from gold to orange, but that was not the life she had been born to. So once again she turned her back upon the distant hill and closed her ears to the song of flight and freedom that the wind sang as it played amongst the hillâs stony ridges. The stream was trickling south by southeast, as near as she could tell. She followed it. Take me to the river, that is all I ask of you. Take me to the river, and I will do the rest.
The hours passed slowly. The stream bent this way and that, and Dany followed, beating time upon her leg with the whip, trying not to think about how far she had to go, or the pounding in her head, or her empty belly. Take one step. Take the next. Another step. Another. What else could she do? - Daenerys X ADWD
Reconnecting with her nature can only be a good thing for Dany. Whether she makes mistakes or not now that she reconnected with her inner dragon at the end of ADWD, we saw with the examples of Jon and Robb that the biggest mistake is ignoring your own nature and identity.