When I turn on either the A/C or heater ,the blower works for a few minutes and stops. I changed out the plug several times and I have to periodically tug the wires on the plug to get the blower to start working. I don't know what else to try.

 

Your problem is with the contact brushes in the blower motor. When you tug on the harness its jolting the blower motor and the brushes are making contact and it starts running. Replace your motor and youll be good to go.