Chris, here is the problem with that propeller always running. The motor is driven by what is called a pulse width modulated circuit which simply switches the power on and off, varying the on-time to control speed. The switching drivers are configured in a circuit called an H-bridge. These are basically just transistors of a certain type that switch fast and consume little power on their own, which are arranged to direct the current in the correct amount and direction for each motor.
Here is the bad news. The only way a motor connected to this type of H-bridge can run without being driven by it's intended circuit, is if the transistors are shorted. They could possibly short out if the motor was stalled for a time and the power was still being switched to the motor through the H-bridge. The motor in this case would draw more current than the transistors in the bridge could handle.
The switching absolutely has to occur normally for the motor to possibly run under control. Your drone has a circuit board which contains the four drivers to the four motors. You will have to replace that board because you can't just fix one of the drivers even though the other three work fine. Those big fat wires from each motor lead directly to the driver board making it easy to identify.