Hi there Becca,
You're not alone. I have social anxiety, and meeting my partners family is a god awful fear of mine. I try to explain it, but he usually tries to reassure me that they'll like me, it's not a big deal, what's the worst that could happen, etc.
His intentions are good, but I think anxiety is complicated for people to understand because it goes beyond reason, so it may appear as stubborn-ness on our part. We may know that realistically we aren't in danger, but our bodies and minds don't.
I'm sorry to hear that you feel that your partner isn't supportive and that your anxiety is a burden on him. What are some of the things that have happened to made you feel this way?