First, let me state my credentials. I was an Account Supervisor for a major auto loan company before I retired. It is absolutely NOT necessary for any dealer to run your credit after your lender has approved your loan. The dealer is simply trying to get you to switch to their preferred lender so they can get more money out of the deal. Don't let them try to pressure you into dropping YOUR lender by saying things like "you lender has a bad reputation," or "the dealership got burned by the lender and refuses to work with them," whatever their excuse may be. If the dealer refuses to work with you once you have secured your own loan, WALK AWAY FROM THE DEAL because your lender is giving the loan to you, not the dealer. Once you make a deal for the vehicle, then you will be responsible for the difference, usually your down payment amount. A lot of dealers will shoot your application for a loan out to many lenders, who will check your credit. Too many inquires on your credit can adversely affect your credit score. (Suggestion: Never apply for an auto loan first if you are trying to buy a house. Secure your home mortgage before you apply for an auto loan.)
There are three separate items you are dealing with when you buy a vehicle from a dealer: the price of the vehicle, your financing, and the value of your current vehicle. We've already talked about your financing, so lets talk about the other two: First, the price of the new vehicle; NEVER go by the MSRP, just remember, it is the SUGGESTED retail price which could be a lot higher than the fair market value of the vehicle. Even if the dealer gives you unbelievable discounts or credits. Tell the dealer you want to see the 'Dealer's Invoice" for the vehicle you are interested in. That invoice is what the dealer paid the manufacturer for the vehicle plus shipping costs. The dealer might include 'options' or 'add ons' on the invoice. That's ok because it gives you a starting point to begin you negoiation with the dealer. Your goal should be to give the dealer a fair profit for the vehicle without being rediculously over charged. Now there are dealers like CARMAX that have a flat price for their vehicles with very little room, if any, for negotiating. They act like a retail store.
Dealers will offer to combine all three elements into the deal as a 'service to you.' But they will simply be trying to control the whole deal to get the maximum amount of money from you. These are three separate items, treat them as such.
Now, the third element, your trade-in. You can trade in your old vehicle in hopes of getting a lower price on the new vehicle. That's ok, if you wish to to do it tha way, but just remember, the dealer will offer you a price lower that you could get by selling it to an individual. You might try Kelly Blue Book at KBB.com to get an idea of the different values of your old vehicle before you decide which way you want to go. Important note: if you own the vehicle outright and sell it to an individual, then any money you get is pure profit for you. If you still owe money on it, your leinholder gets paid first and then you get what is left over, so make sure you sell the vehicle for at least the balance of the loan if you go that route.
THIS IS IN NO WAY LEGAL ADVICE. IT IS SIMPLY BASED ON WHAT I LEARNED AS AN ACCOUNT SUPERVISOR FOR A MAJOR AUTO LOAN COMPANY.