I think you've phrased it very well here--"I have major anxiety over what you think of me. I read into everything and am super insecure. Sometimes I think I need more warmth/connection from you as I never had that in the past. What do you think?"

I think that sounds reasonable, and that your best bet is to frame any request for more warmth in context of your anxiety. I'd be prepared, though, for a "Why does it matter what I think of you? What matters is what you think of you" kind of response. I don't know your T, or what schema therapy is, but I think often T's refrain from giving warmth/reassurance/praise in the face of insecurity because it can (re)create a dynamic where the client is dependent on external forces for approval.

I think one of the harder lessons I had to learn is that not everybody is going to like you, and even people who do start to like you less and less every time you ask them about it.

Good luck!