Has anyone had a car shipped from CA to the East coast

If the car starts easily and runs, you might try https://www.uship.com/. These guys are truckers who are looking to fill blank spots on their trucks. You can choose, enclosed or open. Once you set up a shipment with them, expect to get a call from a trucker from the side of the road saying he can be at your place in x number of hours or minutes. Rates are reasonable, but you will need cash.

Pick up a copy of Hemmings Motor News. There are a lot of shipping companies advertising in there. At some point in time, I've dealt with most of the major shippers. The trucks are clean and enclosed, some hard sided, others have a fancy roll-up tarp. The drivers have been clean, friendly, and know their stuff. Check to see if they will handle inert vehicles, if your is one. Some will handle it, some can't.