Having worked in the auto industry for 20 years, I have seen this happen quite a few times. The most truthful statement you will probably ever get out of a used car salesman is this - if you drive it off the lot and it breaks in half, at least you own both halves.
There is a reason that used cars cost quite a bit less than new ones, and you have stumbled onto it early in your ownership experience. Very sorry to be a bit brutal, but I deal with this all the time, and it is also why most auto experts will advise you to have any used vehicle checked by a reputable third party, preferably somebody you know and trust.
Used vehicles generally are sold as-is, no-warranty unless state or local laws dictate differently. Absent of any remaining factory warranty, or the purchase of a reputable service repair contract, you are pretty much on your own.
With that said, since it was a very recent purchase, hopefully the dealer is willing to stand behind what they sell. If you like the vehicle you purchased and would rather keep it, appeal to the dealer's sense of decency (hard to believe, but most do have one). While they may or may not repair the vehicle at their expense, maybe they would share in the cost of the repair. Or as stated previously, exchange this vehicle for another on the lot. Or, refund any monies paid, cancel the sales contract and move on, since that has apparently been offered. As I've stated in prior posts, COURTESY is key in getting help from a car dealer. They're much more willing to work with you if you are not in the showroom screaming and yelling (not that you've done that). Remember, any help the dealer offers at this point is purely a customer goodwill gesture on their part, because most do want happy customers - their futures depend on it! And as JETX pointed out, get any agreements in writing as soon as possible - dealership personnel can change on a moments notice, and all of a sudden nobody remembers making any deals at all.
Of course, if they are making promises that they are not keeping, most media outlets (newspaper/TV/radio) really seem to enjoy getting their teeth into a story like that - and bottom line is that a car dealer is only as good as his reputation, so they tend to work hard to protect that if at all possible. Be sure to give the dealer every chance to rectify the situation for you, and if you feel you have to use this tactic, be very careful to stick to the facts and not get yourself in hot water by slandering the dealer, because that WILL NOT work in your favor at all in the long run.
Sorry about your troubles, and I do hope it works out in your favor. Chalk it up as a learning experience, and remember to get ANY used car checked before you spend your hard-earned money on it - a good detail and polish can make almost any car look like a million bucks!
