I get anxious at parties too, but have a couple of things I have done. I hope these help you.

One option is to tell the friend you promised to go with (or the birthday friend) that you get anxious at parties. You would have to decide how comfortable you feel telling either one of them. By letting a friend know, they can help you get through the party if they are truly a good friend. It gives you support and someone to be around to help make you feel more at ease. They might even appreciate having you help them get through the party too.

Another option is to decide not to go. You do have the right to decide and never 'have to'. When I do that, I try to make sure the person knows it doesn't have anything to do with them.

A third option is to go to the party and if it turns out awkward, find a way to exit without much notice. I've left parties by quietly telling the person that I wasn't feeling great and was leaving early.