Having moved several times myself, I can give you a few recommendations. Also having worked for U-Store-It in my past, I can tell you that the stories you hear are true....but not with the following companies to the best of my knowledge...
All of these movers protect your items as if they were their own and all of them go outside of Florida to do moves, are licensed and insured.
Even though insurance doesn't help with destroyed items are that are memories, it helps to know they have the coverage.
Little Johns Movers - 352-812-6981;
Suddath - United Van Lines - 407-843-6683 or toll free 800-327-6030 ext 1341;
Move Pros, Darrell McDonald. Office #352-351-5438;
BeltmannGroup - North American Van Lines. 407-295-0636.
Good Luck - If you need some in state or local movers I have some to recommend as well.
