What to do if a moving company lied and put a different cost on the sheet after I signed?

Legally, you don't need to pay more than the mutually agreed upon amount; one party to a contract cannot unilaterally change it, or increase the amount owed by the other party. If you don't pay the additional mount, the moving company would be able to try to sue you for it (it's not clear that they will, but they have the option of doing so); at that point, you'll have the option of defending yourself, but presenting testimony and other evidence showing that the amount agreed to was the $1,900, not the $2,600. Who would win, if it goes to trial, will depend on the balance of evidence, testimony, and credibility. If they try to sue you, you could also go on the offensive--speak with an attorney about whether you can bring a claim against them for consumer fraud; possible contact the police and seek to press charges for fraud; etc. Good luck.