I think whether you are binary or fluid or anything in between, you can get in places like this.
I am personally very binary, very male. I feel very male. I do have enough behaviors that aren't "normal" for a male person that I raise eyebrows occasionally.
When I'm around men, I sometimes feel like a liar, like I should tell them I don't have the "right" genitalia or upbringing. When I'm around women sometimes I feel like a predator, or like I'm giving them the wrong idea about men in general. Like my behavior or opinion as a man isn't quite valid anywhere.
The earlier advice to just tell people about your transition isn't bad but isn't always feasible. I started into this thinking I'd be out to everybody, all the time. Then I encountered situations where it was not safe to be out. Then when I got to reliably passing I got to where I don't know how to bring it up. How do you announce your gender foibles and who you were raised as when nobody asks? It's hard to even bring up sexual orientation when it's not obvious, and it's not like people check your official status on these things when you go places.
Sometimes I miss being a woman, or at least being able to talk about the experience of it. I'm positive that's not strange, and that having internal conflict about your own identity isn't abnormal either. Sometimes I feel like a fake or a failure because I have memories and habits that don't work with who I am right now.
Regarding therapy, you might need to find a therapist you like and then just shoehorn them into dealing with GID. I live in Portland, known (by people who know it) for being progressive and welcoming, and I've still had to do that with some of them. My current therapist is at an LGBT youth center, and well-trained in gender issues, but not better than some of the therapists I've had who knew nothing about the topic before meeting me. Sometimes whether you click with the person and what you need from them is way more important than their prior experience.