About 3 years ago I shipped a Plymouth from Indiana to California. Price was about $1200 and I used Dependable Auto Shippers (DAS). There were some complaints in the reviews, but these guys ship a lot of cars. My car was running, but they most likely ship non-running cars -- but there would be an extra charge.
When you read the contract, it basically says that whatever happens, they are not responsible. Unfortunately, this is probably typical. I didn't have any problems, but it did take longer than the predicted shipping time.