First, read the contract's fine print to see what rights the seller has given themselves. It is my small understanding of contract law is that the company responsible for preparing the contract bears the responsibility of ensuring it prepared correctly and any mistakes they made go against them if the contract is contested. I "think" the cool off period in many states is a consumer protection/purchaser's right, not a seller's right (but I've been wrong before ;-)
Too, it seems to me that once they released the vehicle to you, it is proof they were satisfied with the check and the signed paperwork. Once they put the terms to paper and then agreed to them by signature, usually any verbal agreement contrary to that is void. That's why you always need to read the contract to ensure it matches what was agreed to verbally.
The state's Attorney General's office website may give you some direction on what rights you have under their consumer protection laws. You might even call them to see they can offer assistance without having to engage an attorney.
My biggest concern is the status of the vehicle's title. If you have it in hand, you are in much better shape to have the title transferred to your state and then having the vehicle registered. If they did not provide you with the title at the time of purchase I worry that they will hold on to it and prohibit you from registering the vehicle, at which point you might again consider contacting the Attorney General and if they can't help you, you would really need to engage a good attorney. My recommendation there is to hire on that from a firm known for carrying a big stick and knowing how to use it. One letter from them may do the trick rather than having to go to court.
I'm not an attorney but maybe some of the info will be helpful?
Carolyn