It's a private summer camp. But i'm a public school teacher. Additionally, I want to work to save up money for grad school 
I am pretty sure that a public school would never let you do that. Sometimes, we get short term jobs where we go to another school or do another event. Usually, those are paid. I had one once, but it was only two weeks long.
I think, it is best that you drop it or quit your job or do illegal work. I would recommend dropping it, as it is the best way to stay employeed. Legally, if you are a public school teacher, you aren't allowed to work elsewhere, unless it is another public school/public administration deal.
Good luck to what you choose, but if you need money that bad, I have no idea why you came to Korea.