L.R.
I had a similar experience with my daughter at exactly that age. She was miserable and clingy at birthday parties (even one at her best friend's house next door where she's normally very comfortable) and needed months to adjust to a new preschool. We made the mistake of inviting too many kids to her own birthday party, so she didn't even enjoy that. As a parent, it's very frustrating, stressful, and even awkward (e.g., at other kids' birthday parties). My only advice is to be patient and let her avoid situations that are stressful for her (in our case, we started skipping some birthday parties). I too was worried about social anxiety and how she would cope with kindergarten -- but the year between now and then can make a big difference.
I agree with the other writer's advice that you try to identify a teacher she likes who can help reassure her and ease her into activities. In our daughter's case, she recognized what kinds of situations made her anxious and we reassured her that it was okay to not like certain things (like big parties); I think it helped her to know that it's okay to be who she is. Since that time, she has become much more comfortable in larger groups, including public school classrooms, but she still gets anxious in some situations and that's just her personality. I wish you patience while your daughter matures!