My passion, however, is art, and I try to see as many exhibits as possible.
Yes, your comma placement is correct.
You need a comma before the and because it is a coordinate conjunction joining two independent clauses. The two parts of the sentence are complete sentences on their own, and when independent clauses are joined by a coordinate conjunction (like and, but, or, etc.) you need a comma before the conjunction.
However is an interjection here and also needs to be set off by commas.
I think that's correct, but to reduce the number of commas, maybe you could say: "My passion is art, however, and I try to see as many exhibits as possible."