I talked to someone else at the DOT today, and he says I have a good chance of getting the finance company paid off, and recommended a legal firm. I'm going to be talking to them in the morning. I'm also (somehow) getting the car to a third-party mechanic to find the problem and document it for the complaint I'm filing with the DOT.
The dealer did not offer a CARFAX. The car is listed as previously titled in Illinois, and has repaired cosmetic damage. Whether the damage goes deeper is another question; I didn't see anything obvious when I was under it replacing the starter. Admittedly, I wasn't looking, either, and I couldn't get very far under it.
The car is listed as having been on the lot since September of last year, so I'm figuring it probably isn't flood damage. Collision damage, possibly; severe mechanical damage is also possible.