There is no good way to resolve this. If a government agency makes a mistake and will not voluntarily correct it, you could file a lawsuit in county court seeking a "declaratory judgment" or binding court determination that you do not owe the money; in the lawsuit, you would provide your evidence of when you started working, and if the court agreed, it could officially determine what your state date had been and that you do not owe the unemployment back.But suing a government agency is much more complex than, say, a small claims case; and seeking a declaratory judgment increases the procedural complexity. This could be a very difficult case for a nonlawyer without and attorney, but it almost certainly would not be cost effective to hire and pay for an attorney. It may be best to repay the money.