sorry, Canada here with the free health care.

I can't see gyms covering injuries like that though no matter where they are. It is just so common I would imagine the rates would be through the roof for that sort of insurance.

You may want to have a word with your coach if you are feeling like a punching bag and ask to be matched up a little better to start out, or put in with someone even more experienced who will have more control and help you get comfortable with sparring.