What if you call the moving company and tell them your concerns and request their best crew? I moved 3 times (for work) and the first and third time the movers were a family owned company and they were great. No problems. The second time the timing was not right and we had a local company with day workers and it was not fun! Also have a couple of your quilting buddies come over and help you supervise. Once you explain what you want to them, they can help you supervise because they will understand and can help you keep things in order.
I was told the same thing about packing, they had to re-pack everything I had already packed and put into storage. We did have a few special things that we moved in our cars.