Every interstate move starts with an estimate, and federal rules recognize three kinds. Which one is written on your paperwork decides whether moving day ends with a handshake or a sidewalk negotiation.
The price is the price. If your shipment weighs more than the mover guessed, that is the mover's problem, not yours. It can only change if you add items or services after signing. This is the safest option; many reputable carriers offer it after a proper survey of your home.
The best deal for you: if the actual weight comes in lower, you pay less; if higher, the ceiling holds. Ask for this by name.
A guess, and legally treated as one. Your final bill follows the actual weight and services. The protection you do have: at delivery, the mover cannot demand more than 110% of the estimate before unloading. Anything above that gets billed later, on paper, where you can dispute it.
Fraudulent movers quote an unrealistically low non-binding estimate over the phone, sight unseen, then "discover" on moving day that you have more cubic feet than quoted. With your belongings on their truck, the leverage is theirs.