You can't do that

The problem is that you don't even know what you agreed to.

Besides, if you paid them, you will have no cause of action to sue. First, a civil action can not be treated as duress so that even they threatened to sue, you can not claim that you were paying them only because you were coerced to avoid the court action. Second, if you paid them, the daycare center could claim that the debt was liquidated and your action by paying them was your pre-existing duty and you can not counter-claim afterwards. Even if you can prove that you had an honest and reasonable belief that you did not own that much so that the debt was not liquidated, by paying them your conduct was the avert evidence that there is an accord (settlement) and a satisfaction (you r paying them). In either case, you can not counter-claim afterwards.

BTW, unless a contract falls within statute of frauds, written agreement is not necessary and that is how our society functions. If you borrow something from other people, do you generally reduce the terms to writing?