Twenty years in the military myself, then following a service woman for seven more, I made many moves. Most times the movers took my guns. they wrapped them, boxed them, and inventoried them. Make sure they are on the inventory, each gun by make model and serial number. Not marked on the incividual boxes as that gives the unscruplous warehousemen a convent target. I also have moved them myself by putting them in my vihicle and taking them with me.
I do believe most states do permitt the transportation of privately owned firearms through their state for the purpose of moving. It's easy enough to check with the state police in each state that you will be going through.
Oh yes, I went the gun shop to gun shop thing once, won't do that again. I travel with guns every year while in the states. I go hunting in New Mexico, then Texas, then Tennessee. I drive most times and carry them in the trunk of my car, in the trailer behind a motorcycle, or some years I fly and ship them with me onboard the airplane.
This is from the below mentioned site: "Common sense and caution are important whenever you are traveling with firearms. The way in which your firearms are packed and located in your vehicle are important factors in your compliance with the law."