Can I sue a moving company and win, even though we don't have a written agreement or contract?

Under NY law, the contract to perform moving services would not have to be in writing to be enforceable.  I don't know Kansas law (which might apply) or the law of the state in which the movers are headquartered (which also might be applied) but they are likely to be the same.  Therefore, if the Court believes what you say about your agreement,  you can win.  However, you also have to consider what your damages are.  You may not be able to recover the extra $2000, depending on whether the Court considers you to have been under "economic duress" when you agreed to it.  As for having to wait an additional 2 weeks or so, this is extremely annoying, but have you had to incur any additional expenses (e.g. buying things you needed which should have been delivered already)?  With rare exceptions, it is not enough just to show that you were wronged; you have to show how you were damaged, monetarily, by the wrong, before a money judgment will be entered in your favor.