Here's a list of likely candidates to check:

Big Shot score motor switches (resized).JPG

Can you identify these switches in the machine using the wire colors? If so, you can try blocking each one with paper and see if the motor stops running.

Yes, if you still have the paper labels on the relays, that will identify each relay. Each of the switches on the schematic snippet are labeled with the letter of the relay that they are on. So you use the wire colors to identify which switch is the one shown on the schematic. Tracking down the right switch is half the fun....

The switch labeled "MOTOR 1C" is on the score motor.

Those are the switches that make the score motor run. If any of them are shorted or closed when they shouldn't be, the motor will run continuously. Hopefully, blocking them with paper will help you figure out if one of them is your problem. Then you have to figure out why that switch is staying closed (or shorted) when it shouldn't be. A schematic will be necessary for that - do you have one?