Shop for car financing separately. Your own bank or credit union is a good place to start, but you'll find other lenders, as well. Ask your car insurer, and it may also be worthwhile for you to join AAA, who will certainly have resources for each aspect of this.
When you walk into a car dealer, they are trying to wrap together (up to) four transactions into one, so as to maximize confusion as well as profit, of course. You need to separate these, mentally at least, and ideally use parties who give you a good deal for each component:
1. Selling your old car. (Hopefully, your insurance company took care of totaling that.)
2. Closing up the old car loan.
3. Buying your new car.
4. Financing your new car.
You have a good head on your shoulders. Some people will try to prey on emotions. Do not be afraid to get up and walk out. "Perfect" deals are like buses; another one will appear in twenty minutes.