What are my rights if I need to prove a cash transaction without a receipt?

You may be, to use your language, "totally screwed," unfortunately. That's not to say that you can't take legal action if necessary--e.g. sue the dealer for the $3,500--and rely on your own testimony, but the problem is that with only your own testimony to rely on, you have an uphill fight. The dealer will likely have more people (e.g. finance manager, sales rep, bookkeeper, etc.) to testify that you did not give them the money; also, the mere fact that you cannot produce a receipt will act against you, since the dealership will likely testify that they always give out receipts (so if you don't have one, you didn't pay) and also it is most people's (which includes the judge and jury's) common sense observation that if someone is paying $3,500, he or she would probably either do so by cash or credit card (so the payment can be proven) or else make sure he or she gets a receipt. So you may legally try to take action, but it it unlikely, from what you write, that you can prove that you paid.