Jamie...
Check the wiper motor switch, since if it doesn't have a good ground, then the motor will not work. The motor itself is also isolated from the ground system of the car, and the switch provides the path to ground. You will also find that there is a ground strap on the motor that grounds a section of the motor to the firewall. It is very convoluted as to exactly how they work, and when they stop working, there is not much that the average person can do to fix them other than to remove and replace.
When you turn the switch to the on position, it provides a ground to the wiper motor, and on the inside of the motor, there are point contacts that work off of a cam that keeps the motor activated. When you turn the switch to the off position, these points continue to feed power to the wiper motor until the wiper arms are in the correct parked position. This is where the wiper motors usually go bad, and it needs to be serviced by someone that understands the circuitry, and knows how to repair them. When you add into the mix, the windshield washer system, it makes it further complicated. I know that sending them out to have them rebuilt is expensive, but this is one of those times, that it is well worth the cost, because of the complexity of the units. Most times when you find a used one on a car, it is also not working properly.