How does an owner operator hold a shipper legally to a verbal shipping rate?

by Kenny Kay
(Hallsville, Tx 75602)

As an owner operator, how do I legally collect from a shipper " the rate" quoted over the phone? In other words, I haul a load of cows from point "A" to point "b". All animals walk off un-skaved.

The shipper say, " I am not paying the rate quoted, because I could have gotten the load hauled cheaper for a dollar less a mile!" how do I legally receive payment for the original rate?