1983 Chevrolet Truck Wiper motor

When you put it on the other truck, are you switching just the motor or the mounting plate too? If you're not switching the mounting plate, check for voltage on the case of the motor when the wiper switch is turned on. If you find voltage there, check the ground strap between that mounting plate and the mounting bolt.

If there's no voltage on the case, (which there should not be), check for two voltages on the wires to the motor. One is to keep the motor running after you turn off the switch, until it parks, and the other one makes it run when you turn on the switch. If that second voltage is missing, there is most likely a break in a wire between the switch and the motor.