You are incorrect. Contract law is old and comes from the common law. In all states, if each side promises to do something, then a contract has been formed. Written contracts are much easier to enforce, because it is hard to prove both the existence and the terms of an oral contract. You seem to talking about unilateral contracts, which are binding but actually quite rare, where only one party promises to perform.

I know you have a question about something that happened, but you asked for a rule of law instead. What a lawyer needs to help you is a list of the facts, so he can tell you what law applies and why.