I had a similar problem when purchasing a truck about 15 years ago. I traded in my old truck on a new one. Old truck and title were given to dealer and I took possession of new truck with signed contract including financing arranged through dealer. A few days later, dealer called and said down payment had changed from $50.00 to $500, but they knocked off a 0.5% on interest. Worked out OK, so I agreed, but nothing in writing. A few days later, dealer called again and said down payment was now $5000.00 and no change in interest
!! I told them that was BS and would just go with the original deal and the only one that was in writing. Dealer said they couldn't do that. I showed up at the dealership, turned in the keys to the new truck and said I wanted my old one back since they couldn't honor the signed contract. They almost left brown stains in their chairs. They had already sold my old truck. I told them it was no big deal and I'd just call my lawyer for a ride home from the dealership and they were not to contact me in any way and all future communication would be through my attorney. I left driving my new truck on the original deal (the one they said couldn't be done).
As a side note, that dealer went bankrupt about a year later. Owner started a new dealership and it went belly up in about three years. Haven't heard anything else about the owner since. I have heard many horror stories about that dealership since my dealings with them, including people calling the police on the dealer when they would take a car into the service area to get it inspected for trade allowance and then refuse to return the customer's vehicle to them
. I've heard that story from several police officers that answered the calls and from several would be customers (most who didn't know each other), so I consider that to be a reliable story about that dealer's practices!!
After that experience, all items relating to the deal are done before they get my vehicle and I leave with the new one. None of this stuff about take the new one home and we'll finish tomorrow . . .
It all depends on the integrity of the person or institution you're dealing with.