you need the resistor first off, when you push the foot switch you're bypassing the resistor, and it starts, when you let off the starter, the bypass is opened, but a bad resistor doesn't provide the current to the coil to keep running, that's the classic symptom .... the starter is a different matter, a foot start has no 'solenoid', the spinning is very possibly just from inactivity, probably go away if you use it regular
Bill