Just be careful now that you are
teaching illegally. You can be easily replaced and the schools know this. In the end, if you are caught, then you will be the one blacklisted in South Korea with the visa office. I would be very carefully teaching for extended lengths without a work permit. If your current agent cannot give you a more definite reply other than "we are working on it", then time to move.
Yes, if you don't have a visa, then the school really can't hold you put. Start looking for work elsewhere on your own and then when you find something, tell your current agent you are handing in your resignation.
You've just arrived to South Korea, so you should be getting settled asap and enjoying your time rather than fighting a losing battle.