That's true under certain circumstances and in certain states, but they might have a hard time making those arguments while simultaneously asserting rights under a written contract.

I am not aware of any way to compel them to produce the contract until suit has been filed, which really bites.

Consider stepping up your efforts to get them to show you the contract. You might want to start daily phoning, emailing, writing, and/or, if feasible, stopping by their office. Being a squeaky wheel often works.