It sounds like you might be feeling there is a hidden message or motive to her questions. I think this speaks to trust which you and she will work on again. Tell her how you feel about her asking. It's could be a great therapy discussion

Have you asked her the same question about working together?

It sounds like you want to work with her and that's why you went back to her. That could give you your answer to her question about that.

When I met with my T the first time she said she meets with a person several times to see if she and the person could work well together. I was a wreck the first several sessions wondering if she was evaluating me in that way and finally I asked if she had decided! She had decided right off that we would work together well (and I think that was a way to give me room and time to see what I thought about working with her and that it would be okay if I chose someone else) She was sorry I had been feeling so anxious about it.

What I want out of therapy is similar: to feel better, to feel like I have someone in my corner, to learn and grow.

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what I want out of therapy, deep down is something that I want from her, I want her to be there for me and care about me.

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That's your answer here, and it's okay to tell her this when you talk to her.

I hope you'll keep going and talking about these things with her. You're just getting started on this round of therapy and the awkwardness will settle down and it will become the safe and comforting place of your wishes.