Yeah, cross country driving is better suited for a diesel. Might wanna look on diesel specific forums, but I'll tell ya what I know. The 7.3 Fords will run forever, but the trans isn't the strongest. Kinda slow as well, but like I said. They'll run forever, and you can get em for cheap. The newer 6.0 isn't terrible. But it's got a bad rep and I'd never buy one. The newest 6.7 is an awesome motor from what I've heard. Lots of power, quiet, and sip fuel.
As for Dodge and Chevy, I don't know much. Cummins is always a safe bet, as for the Chevys.... I know nothing. I know Dodge put a stick shift behind alot of those cummins. So that's good if you can drive a stick.
You could get a gas motor... but I don't see the point. No better gas mileage, if loaded you'll get worse.
A F-150 wouldn't be a bad choice if you aren't making this trip but a few times a year and only need the truck for the trip. But realistically how much weight are you going to be carrying? Factor in kids, wife, tools, food, clothes, everything. It adds up quick.
The new F-150 would be fine, but those are expensive. Older F-150s might not be happy if you're carrying 8,000+lbs