Move it empty. Mine was just moved due to a house fire - everything had to be cleared out of the house. The company that did the clearing, deodorizing, etc moved it full. None of the drawers were overloaded and the drawers were locked when they left.
It destroyed the drawer slides.
These boxes are designed to be moved across shops, not down the road in a truck. A full drawer of tools bouncing up and down in a truck for thousands of miles will damage the drawers, slides or something. Heck, mine was moved less than 100 miles and it was damaged (it was not a cheapy 21 gauge frame box either, but a 30+ year old made in USA heavy box).
Yes, it's a pain in the **** to empty and pack everything. However, think about the cost of that tool box. On the plus side, they bought me a new one, but you'll be buying / fixing yours. It's worth the time to remove the contents.