Before answering the main question, I want to say some important things.
First of all, as someone who was on similar path, take it slow and easy. I've been there and it can get really overwhelming, especially with all of the confusions ans uncertainties, especially when you think you figured out your spiritual path and then something new appears that breaks beliefs you previously build.
Second, if it is the King reaching out and you are uncomfortable or that's not for you in any way, you can always say no. Deity reaching out doesn't meant you have to build a relationship with them, you can always reject or ask for more time or any other way it would be comfortable for you to resolve.
So, now, the main question itself, how does one a Deity/spirit/something is reaching out? The answer, I'm afraid, not as clear as you'd like, because I'm not you, so I can't say for sure what you had experienced.
I would recommend, at first, to read this post from @vesse-moricaleo-lusifero, as well as all the links that are provided in the post itself. It is about Lucifer mostly, but all the key points are there.
The reason I said to take it slow and take your time is precisely that. To see, if it is the King reaching out. Several days is quite a little time to say for sure. Again, I am not you, I don't know your whole story. But the main points of a Deity or anything reaching out are: consistency and same pattern. If, for now, it's been some time and those pattern that you noticed are still present, it very much could be it!
To add more, I can provide my own experience, as well as DM you a TKIY worship Discord server, where you can lurk, ask around and see, if it's something for you or not.
My own experience with the King is such: I, actually, at first, wanted to just support a friend that got into worship of the King in Yellow, but, after some time of lurking and supporting as much as I could, I began to feel A Thing. A presence, then patterns. Quite a distinct ones, both, so I knew it wasn't one of the Fears. I didn't jumped right into it, I let it marinate for a while, observed it. I let my own fears, emotions and insecurities to calm down, so I could actually see clear what is happening. Then I picked up the book, the original The King In Yellow by Robert Chambers, and so began my research part. With getting actually understanding the King, I got to understand what those patterns were, with being in a community, I got to experience the way the King shows himself for others and how he shows himself to me. The final point when the decision was made to add the King to my pantheon was, actually, pretty recently, about a month ago, since all that time I was observing. Now I have the same process about Nyarlathotep, waiting, until my emotions calm down, and then diving into the research.
I hope I could help in any way. You can ask my friends and fellow King worshippers @portal-two and @web-wound about it as well or lurk around their blog. Take care, mutual. :]