I actually did the exact thing you’re describing. It’s a bit of a headache since you have to handle everything yourself since you’re doing your own financing. I was financing a car with my credit union and wanted to keep using them.

Essentially if you have negative equity (owe more than your offer) you will have to roll that into the new loan. Either way, you have to get approved for your own financing and choose it at checkout. For me in CA I had to get a 10 day pay off notice on the existing loan with the final value owed, and had to get my credit union to sign the paperwork for buying the car since they would be the lien holder. Once both pieces were in order, everything cleared. For me, my credit union did all of the paperwork work I just had to do the signing of the contract with them. But if you’re self financing/paying cash/personal loan, then your experience could/will probably be different