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We did our loan through NFCU last summer.
We asked for a certain amount without having the car negotiated yet. We basically stated our max and then we were approved for it at the lowest possible rate they advertised at the time.
We went to the dealership and drove the car. We didn't talk numbers, we wanted to sleep on it. The next day the check came in by Fedex and we went to negotiate the car. We took the check and car documents...that's it, the dealership doesn't need any paystubs, etc because we had our financing in place. We told them up front that we had gone through NFCU, they asked what rate we got then said "that's awesome. We can't beat that" so then we negotiated the deal and went through that process. Out of there in about an hour and a half because they had to clean the car up.
Best car deal we have ever done. It was a CPO Lexus.
You should be able to go to any dealership. You will basically be taking a blank check in with you!
It was seriously the smoothest transaction ever. No haggling back and forth with their financing people and our credit getting dingged a bunch of times.