Some companies may ask that you put down a small deposit when scheduling a local move but if you are moving out of state, you should NEVER put down any money up front. You will be required to pay for your move before anything is unloaded from the truck but not before anything is loaded.
Long distance moves are based on weight. Your estimated cost should be based on an educated guesstimated weight that is calculated by taking a visual inventory of everything that is being moved. If your move talks to you about how much space you need and not about the overall weight, keep looking for a mover.
Also, if your mover takes only cash or money order, you need to find another mover. With internet credit card payment processing services linked to smart phones, there is no reason that even the smallest mover cannot take a payment via credit card.