It has been a very long time since I have been in a relationship...and unfortunately it predates my anxiety. However, I know how you feel about seeing your friends starting relationships and doing "couples things". All my good friends from home are married and have kids, but I'm still single. Sometimes I'm okay with that....sometimes I'm not.
However, I worry that if I do have a relationship, that my anxiety would also hinder it. While I'm not agoraphobic, I really don't like going out to parties, clubs, etc. I'm much more comfortable with just a few people, sitting around chatting or watching a movie. I'm NOT a social person. Now, I simply don't go out and it can be very lonely.
I hope you don't let your anxiety hold you back. If you are honest and explain to the person you are dating that you have anxiety, if they really care about you, it shouldn't matter to them.