Can a car dealership let you drive off with two cars and then a week later tell you that you weren't approved and that you need to bring back the cars?

It depends upon your state, and the contract you signed. The retail installment contract (long form) will have a paragraph/provision for "seller's right to cancel" and spell out how and when they must contact you. If they miss the time period listed in this provision, then they can not cancel.

Also, if they try to get you to put more money down, and/or change the terms of the contract, you need to make sure the contract shows the new date and not the old one (it's fraud if they backdate).

One way around this is to improve your credit - the dealership approves your financing based on their belief that they can get someone to purchase your loan. If they can't find a willing bank, they need to get their car back. If none of their banks will buy your loan, it means your credit is that bad.

Pay your bills and lower your debt.