I have a 220V AC unit with a 15A plug, but the socket in the wall is a 220V 20A socket. What would be the best way to go about fixing this problem? Should I get a new breaker for the box and a new socket for the wall?

Yes, Change the breaker to a 15 amp and the outlet to a 15 amp. The wire should be adequate.

I've looked all over the place for a 15A 220V circuit breaker and can't seem to find one. Do they exist? Or do I need to use a 20A Breaker with the 15A wall plug?

There is no 220 volt breaker anymore, they are 240 volt. I just googled one and found it.