All three times I've done long-distance moves I was able to carry at least some of my guitars with me. The least-valuable ones went in the moving truck, in standard guitar cases, and I made sure they were padded with towels/bubble wrap on the inside as well, and that that movers put them in places that would support them well. Never any issues. Definitely wouldn't just use gig bags - even the ones in my SUV got new cases if needed because it was pretty packed too. I also padded my amps in boxes, pulled the tubes, and put wood or at least cardboard over the grill so the speakers didn't get punctured.
Also, make sure that you can lock the cases, or at least wrap a bunch of tape around the cases and across the latches. That way they won't just pop open by mistake, and you might also be able to tell if the movers mess with them.
Depending on your move, it's not always as simple as "into truck at your old house > out of truck at your new house" - do your research first. Movers often give wide delivery date estimates - not because they don't know how long it takes to drive from point A to point B - but because many do pickups and drop-offs on the way to maximize their efficiency, which means they may rearrange your stuff in the truck. Many also will take your stuff to warehouses along the way, remove it all, let it sit for a couple of days, and then reload into a different truck that may still make other stops along the way. Our last move we almost got screwed on that one, fortunately we realized it and found someone local to Utah who drove straight to our new house in CO. It helped that it's a one-day drive but cross-country is a different deal. It definitely helps if you're able to fill the truck - then they're more likely to just take your stuff straight there.
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