Since you will not be going through New York or New Jersey most of the other states you'll visit should not cause any problems unless you give the troopers a reason to stop you. Remember on COPS they do all the stops because a tail/brake light is out, or a rolling stop or other MINOR issue and then bang. If you plan to stop for the night I would make sure the car/truck you are driving is disabled so it will not run, no matter what or take everything inside with you every night.
I am surprised no one gave any actual references so here are some I had put together for some relatives who were moving a while back.
THR
TruckVault - Gun Transport Laws by State
Cross-country moves with guns, How to do it legally? - Calguns.net
https://www.nraila.org/gun-laws.aspx