Get checked out.... You may feel fine, you may be fine, but at the least it is documented in case something should come up down the line.... Their insurance company shoudl pick up that tab...
Document every gee darned thing that has anything to do with your car, you, and the insurance company.. If they call just to say "Hi" DOCUMENT IT! It's sort of a pain but it'll cover your butt..
I know rules vary wildly from state to state so you may need to read up on some of that.. Have they had THEIR adjuster look at the car within six business days of the accident? If not there's a slip up on their part..
There a matter of where the car is at.. If it is on your property and they insist on writing is off, the owe you storage from the day of the accident till the car leaves.. Generally you can offer to forget the storage fees (on average $25/day)in turn for keeping the car..
Another thing to keep in mind is filing complaint's against the company.. THe Better Business Bureau is a good one to start.. Your state's attorney general, or insurance dept (whichever you have) is the best..
In PA when a company goes to renew it's liscense to sell insurance in this state all complaints are reviewed in the renewal process.. Insurance companies don;t want any complaints as far as this go, let alone unresolved ones... Just had a go round similar to your's over my niece's crunched 300SD.. 10 days after filing the complaint (typical time period for it to be processed) it was resolved with satisfactory results...
Good luck!