I would not go to a dealer to trade it, despite the fact that most dealers are relatively clueless. I'd try for a private sale because you will almost always get more for it that way unless it is a common car in your area.
Probably should have gone through the other insurance company too, because then you can just fine the best shop in the area and take it there. You can do it through yours also, but it is usually better to not make a claim on your insurance even if it wasn't your fault.
Doesn't matter what the insurance company wants, you have the right to have your car returned to the condition that it was in prior to the accident. I got hit and had to go to the doctor, so they towed my car to some random shop that I did not approve. When I went to pick it up, there was a big run in the hood and I told them I would NOT accept that. I called the insurance company while I was standing there with the owner and told them not to pay them until they fixed it...and they did.