You're answer is going to depend on what type of move you're trying to arrange. Local moves, intrastate moves and interstate moves can all have different regulatory requirement depending on where your moving from an to.
On shipments involving interstate commerce, the Federal Consumer Protection Regulations included in
49 CFR 375 state that the bill of lading is the contract and the customer is free to change anything up the point that they sign it. Many states have the same regulatory requirements. Since you mentioned PUC, I suspect this is an intrastate move.
Sounds like you're dealing with one of the web vultures who prey on inexperienced and uninformed internet shoppers.
I'm not an attorney but I think most transportation lawyers will tell you they don't have a leg to stand on.
When you're moving, you best bet is find three reputable movers and get them to provide you with in home estimates. Sound like you already found one.
If you are moving interstate, I suggest your download read
Your Rights and Responsibilities When You Move at the Federal Motor Carrier Safety Administration's
ProtectYourMove.gov website.
You might also want to check out
How to Find a Professional Mover.