Tips on Buying Cars: Dealership backing out of lease, time crunch, audi a6

Question
My soon-to-be husband and I went to get him a new car a couple of weeks ago. I had a 2000 Audi A6 that I was upside down in the loan and he thought leasing a car and offering that as a trade would relieve the debt-to-value problem. I agreed and traded in my car for a lease that would be solely in his name. I signed all of the paperwork to release the title to the dealership and my betrothed signed all of the lease papers. His credit is not as good as mine, but is in the high 600s so nothing should have prohibited this lease from going through. Obviously, the dealership agreed because they had him sign, took the car and away we drove in the new leased car.
Now they're trying to say that since he has not had a car loan on his credit report spanning the last 10 years, that I need to cosign on the lease. I don't want to do this because I feel like I have enough new credit lately and don't feel that his lack of a car loan should be detrimental to this agreement. That is one battle.
The one I'm more concerned with is that, since their leasing company has not approved this lease yet, they have not paid off my car and I have a payment coming up in the next few day on a car that I don't feel I legally own anymore.
Any help you can give on either or both of these related issues would be very much appreciated. I'm in a bit of a time crunch, too, since I have to either bite the bullet and write my name on a lease and nastygram to the dealership or give back this car and take mine back in return.

Answer
First and foremost: protect your own credit rating and make the payment that's due on the A6.  If you've overpaid since the dealer obtained a closeout figure on the Audi, you'll get the difference back once they've received the payoff from that dealer.  If you're concerned this may not happen, I see no reason why the dealer can't produce a letter for you stating as such.

Okay, that was the easy part.  As for the rest, sit down and read the lease contract, word for word (you did receive a copy of the contract, right?).  Is there any clause that states the contract is subject to bank approval, even after you've taken delivery of the car?  If you have a car, and a signed contract without any clause like that, then I'd point that out to the dealer and ask, "what's the problem?"  In short, ask them to show you where it states you need to sign a contract that's already in force, on a car you already have in your possession.  Such "spot deliveries" are legal in most states (but not here in New York), but need to be handled in a manner much more carefully than what your dealer did.  You just might have them over the barrel.  If there's anything that's not clear, please ask a follow-up question.  If you need to act immediately, then I can provide you with my cel phone number, but in a more private manner than this open forum.