I'd really recommend AGAINST an open transport. My last move, I had my truck transported (open carrier) from central Florida to L.A. When I finally got the truck about 2 weeks later, it was absolutely FILTHY. Some of the stains in the paint never came out. It looked like it had been sitting in every grimy, filthy place between Florida & California... dirt, grime, bugs, grease,
oil, you name it!
In past moves, 2 of my vehicles were transported in the moving van along with my things, while I drove the 3rd. This last time, had been trying to sell the Lincoln (but didn't), so ended up driving it (most comfortable ride) & the truck ended up on an open carrier.
I feel for ya tho... have made that drive several times. None of the cross-country routes are very good. I like I-40 the best. I-10 is pretty through Arizona & New Mexico, but is the pits going through Texas.
If you value your car, I'd do one of 3 things...
-have it shipped in closed transport
-drive it both ways
-leave it at home & take a rental one or both ways.
... my 2 cents worth.