Connie,

You are only charged by weight if you are moving long distance. For example, if you need to move within the same town you wil be charged by hour, and if your move is cross-country you will be charged by weight.

Some moves you could choose: for example, if you are moving across the state, and your new home is few hours driving from your old one (lets say for example if you were moving from Philadelphia to Pittsburgh). 

In those cases you would have the choice of either hire a local mover who will drive your items straight to the destination, and charge you for the time it took to complete the whole move (including their return back), or hire long distance movers who will store your items in a transitional storage and haul later on a long distance truck (longer, but sometimes much cheaper).