I wouldn't do the resignation thing. You don't know how long you may be out of work, and you need the benefits more than you need the reference right now. You will survive without the reference. They will have to confirm that you worked there, and save whatever you received that put you in that "either-or" position. That will show any future employer WHY you cannot get a reference.
Besides, you can always get parents to write you letters saying you were a great teacher. I would reach out to some of them the minute you are no longer employed (in case it's considered unethical or anything while you are still working).
What I was suggesting about contract employment (aka "temping") is that you consider any job for which you would be qualified--not just teaching. Some of those jobs pay pretty well, because there are no benefits to factor in. Worth considering, anyway.