I have over the years arranged a couple dozen interstate moves for employee relocations. The protocol I had implemented was to have 3 estimates all from National moving companies. Companies like Mayflower, United Van Lines, North American, etc. These are movers that will operate point to point, with their own equipment. Make sure they have sufficient insurance, ask for the insurance certificate, and to name you as “First Insured.” Take out any additional insurance as you feel prudent.
The national companies will not be the cheapest. Some of the lower priced quotes you will receive from other companies will not be the ones that actually move your items. These companies act as consolidators, and ultimately your items are sectioned off on a moving truck with other peoples goods. Among other things this means that your items are being double handled which doubles the chance of damage occurring. The arrival dates on your items can vary wildly depending on a number of factors. Additionally, should there be an insurance claim you can end up with the finger pointing game.