We contacted and interviewed several companies, who came up with various estimates.   Most were pretty much the same, except for one company that was very high.   When I asked their representative about it, he said that It was a binding estimate.   He also said that few movers will give you a binding estimate, that most will only give you a non-binding estimate, and that whoever we went with, I should insist on a binding estimate.  In one article I read about moving, I remembered the advice that you should include on the contract the following phrase:

Customer will pay the estimate or actual cost, whichever is less.

After we had chosen United Van Lines, I asked their representative if I could include that phrase on the contract.  She said that it would be OK for me to add it in and sign the contract, so I did.  I dropped the contract off on the day before Christmas Eve(or Christmas Eve, I'm not sure), when I went by their warehouse to pick up some used boxes to pack up the stuff that we were going to pack ourselves (this was on a Tuesday or Wednesday).  We packed up some things on Thursday, and the United folks were going to show up on Friday (the day after Christmas) to do their packing.  We wanted them to pack up the valuables (dishes, silver...) so that it could be insured.