I recently moved and used a moving company and was very pleased, however, I did not know that they were going to disassemble furniture so be prepared for the potential for missing parts. When I saw what they were doing, I set up a box specifically for "parts" and made sure they wrapped the parts for each furniture in their own package....then, I wrote on the packages what piece of furniture the parts go to. For example, they disassembled a tri-level computer desk so there were screws and bolts and what not. They took legs off of living room furniture, removed glass doors from bookcases... My baker's rack had so many parts I doubt I would have been able to reassemble if not for this plan. They disassemble in order to make it easier to stack and pack in trucks. It's amazing how they do it but a little unnerving.
I had 3 estimates and the one we selected was within 100 lbs of estimate...so some are pretty good at estimates. In terms of keeping things in drawers etc. I emptied them since my stuff was being delivered directly to a storage unit until my new home was built; however, I did ask about this and they said I could leave unbreakable stuff in the drawers if I wanted to. My husband has a lot of tools and one fairly large tool box.....super heavy.....We thought we'd have to remove everything but they loaded it in truck fully loaded since it was on wheels......that thing was heavy...
Good luck on your move!!