I would not walk out of such a deal, a promise is a promise, even if I had made a mistake with the price. Who in their right mind would sell their
reputation for a $1000?
A year or so ago I colleague that is into photography told me he had bought a new camera and was selling the old one. The old one was a very nice professional level Nikon. For the low price he named I immediately agreed to buy it, I even paid for it the same day. He was supposed to bring he camera the next day, but instead of bringing the camera he said be would like to back out of the deal because he could get a lot more if he took the camera to his camera shop in exchange for some lenses.
I had no problem with that for at least two reasons: he named the low price before he knew the real market value (I did not know the market value either, but I recognized a good deal), and he asked me for a permission to back out or pay the real market value. I could have stood my ground and made him sell the camera for the low price he promised, but that would not have been right.
As to buying cars privately...
I bought a truck recently from a private seller for a very, very nice price. When we shook hands to seal the deal I paid $500 upfront and requested a receipt. I took a calculated risk giving $500 to a stranger but now it would have been a lot more awkward for the seller to pull out of the deal if someone else offered more money. I paid the rest the next day when the banks opened.