I'm no therapist, but I get the feeling there is more bothering you than just the name thing. Sometimes putting the feelings that make us feel uncomfortable into a neat little box, in this case names, is a way of avoiding a bigger problem. if you have a therapist, that is something to bring up. If you don't have one, get one.
Just a hint, the one thing I never, ever tell anyone is the name I had before I transitioned. That saves me from thinking about it and others from asking about it. If I were to meet a trans woman, I wouldn't want to know her birth name either. Some things are better off left behind.
Even if you feel uncomfortable going into the men's room, go in with the attitude that you belong there as much as the cis guys. If they don't like it, tough. Sometimes acting confident turns into being confident. I've seen some cis guys that i really wasn't sure they were guys or girls. Do what you can to keep your appearance as masculine as you can.
sam1234