Driving a U-Haul across country isn't something I really want to do, but I'd do it if I had to. The wife would want to hire movers for most things, but we'd also probably get rid of a bunch of stuff that wasn't absolutely needed (sigh, goodbye classic computer collection). Still, it's going to be a pricey prospect no matter what we do.
I wouldn't expect a potential employer to pay for the entire move, but I'd hope for some sort of moving stipend. I actually never thought of telling a company I was actively trying to move while I was interviewing, that might help. Still, how many companies are even willing to fly you out for an interview unless you're something special? Back when I got out of college in 2000, I had multiple companies fly me out for interviews. Those days are long gone I'm afraid.