I love the 13th doctor (and to a connected extent, 14) because she's so obviously fucked up, so clearly not handling things well, and deeply wishes she could
To go from 12, this coarse and broken man who was softened up by companionship and duty to his moral philosophy
To being 13?? Is so jarring, but not at all surprising. 11 to 12 was the doctor sort of going Mask Off, rejecting a facade of youth and whimsy in favor of a more honest and alien approach. He's not interested in being/looking too human, he confronts problems head on, and he's very raw. His emotions are concentrated, to the point that at the end of his life he nearly refuses to regenerate as to stop this cycle of suffering.
But due to the show going on, he becomes 13 who becomes a completely different beast.
Her first season is sort of intentionally vague. She never talks about where she's from, or opens up. The Tardis team at the time just sort of roll with it, for a long time just being along for the ride. But when pressed on it, she just shuts down. And in a weird way, that is a Hallmark of her character.
13 is mainly characterized by a sort of...performative enthusiasm. She constantly shouts out about how she loves things, makes reference to things in pop culture that she thinks her Fam would like. She comes across like someone really trying to fit in, business as usual. Like a divorced mom trying to connect to the kids with Ed Sheeran and Minecraft, desperately distracting herself from the trauma of the ongoing separation. But the mask falls under pressure.
Whenever confronted with something sufficiently out of her wheelhouse, she shuts down, either freezing up completely or getting angry and cold towards those close to her. The Fam asking about her home get empty responses or obvious lies. When real danger strikes she spends most of it grimacing, pleading a case to stop the madness.
Nearly the entire Timeless Children storyline is her cooped up, utterly silent as The Master infodumps troubling information to her. She comes across as someone who is utterly numb, dissociating when confronted with trauma that is too personal. Sitting like someone who has a long list of traumas that are piling up.
During series 13, the entire subplot with Yaz and romance is sort of brushed away by her in a way that indicates something is more wrong than she ever lets on. She retreads beats of loss, of Rose and River, sort of positioning Yaz as someone she just can't make decisions like that with. But through that are genuine moments of affection that seem tinged with an unseen weight.
The 13th doctor comes across as an incarnation acting as a coping mechanism, but a bad one. The Doctor putting on a brave face to keep doing what they always do, even when they don't really want to. They feel like 12's instructions for the next doctor were being handed to someone who isn't in the right mind to handle it. She's someone who puts on a brave excited face seeking the wonder in the world, but deep down is too distracted by all the constant baggage to really enjoy it.
She's got this superficial wanderlust as a means of trying to find the joy in the universe, but eons of things have piled on to make it hard to genuinely smile at them. She can see beautiful things, wonderful sights: the Solitract, Yaz's grandma getting married, all the teams harmless side trips off screen. But the dangers they end up facing and the constant horrors remind her of the things she's avoiding.
She's deeply emotionally unavailable in any meaningful way because she's shifted the inhuman timelord nature completely into her capacity to detach herself from anything personal. She's a new mask, a new coping mechanism for the doctor. Just like 11 being a youthful face to conceal the years he's had to endure, 13 is an enthusiastic face to chase the glimmers of the universe that make life seem worth living. Like a walking talking "manifestation/positive affirmation" incarnation. She tries to speak optimism into the universe, as more Cybermen, The Master, the Flux, etc all hang over her.
It's honestly sort of revealing that she's zapped and dying, and the first thing she does is put on a brave face for Yaz, take her for ice cream, and then leave her. She goes off on her own to confront death, and almost seems relieved to hand it off.
She turns into 14 and all of those problems, all that baggage is /HIS/ now.
14 adopts 10's face and vibe, but he also keeps some of the mannerisms 13 had, voicing how much he loved people when 10 refused to tell anyone how he felt. But initially, that's the point! 14 is still sort of puttering around with all that baggage, inherited from all the others. Able to be more emotionally frank about those close to him but unable to truly confront all the things that haunt him. Donna recognizing that he may have returned to an old face as a means of coming home and resting is the closest we get to see on screen of him really getting to rest and deal with all the trauma and suffering they've experienced.
Say what you want about RTD2 and Chib's, but I think it's interesting how they characterized 13 and built off of her in a sort of "Progress isn't linear" kind of way. Like sure, 11 saved his people from the war, but does that undo the trauma?
Imagine you've been put through hell. Fought in a terrible war. Your childhood bestie keeps coming back from the dead to kill you. You've had to deal with losing all your family. You get a fresh start and get to hang with new friends to distract you from the horrors of the past, just to find out the universe is being erased by dark matter and it's all your fault. I would probably shut down too