My advice, don't ship it. I've heard too many nightmare stories about cars getting dropped off car carriers or things leaking onto their paint from the car above. Plus it is wicked expensive. I had a car shipping from Oklahoma to NY and it cost me 1200 bucks. The car was a parts car so I paid more to have it shipped than I did for the car. It did not matter to me that it got stuff on it when it was on the carrier because I was only interested in the drive train and few other sundries from it. But it wasn't cheap and the shipping company was very shady. The driver was contracted, he didn't work for them directly and I have no idea what insurance if any there was on the whole rigamarole, not that I cared.
I would say drive it back yourself. If those mountain roads are really that bad, rent a panel truck, chains and a car trailer and trailer it back yourself. It's not worth having the paint destroyed on your new used car or having the jerk with the car carrier drop it or go off the road trying to get it to you. Trying to make a claim with those guys insurance if they have any would be a nightmare.