Here's what I would have done before replacing anything:
Note: The upper pop bumper on Pinbot is driven by special solenoid #6. These instructions are for that circuit but the logic applies to all the special solenoid drivers.
Start by disconnecting the coil that's stuck on when in a game, if you have not already. Put a meter across the wires going to the coil. It's better to troubleshoot with a meter than a coil.
The solenoid only energizes when you start a game so we know it's not a shorted driver or pre-driver transistor. If either of those were shorted the coil would energize as soon as you turned the game on. The first thing to look at is the stuff hanging off the switch side of the NOR gate in the pre-driver circuit.
Logically you could split the circuit in half at the NOR gate. One side comes from the switch and one side from the CPU via a PIA and some logic combining it with the blanking signal to basically "turn on" the circuit. The switch side also has a PIA output connected to it so the CPU can drive the circuit when it feels like it (like test mode). The CPU "turn on" side seems to be working because the coil is only energized when you start a game so the switch side seems shorted.
The first thing to do is eliminate the switch as the source of the problem. Remove the connector coming from the switches. If your problem goes away you know either the switch is stuck closed or there is a playfield wiring problem.
If that doesn't solve the problem I would look at C74 and ZR7. If either of those shorted you would see the problem you're having. Check pin 5 of U50 with a logic probe. It should be high. I'm guessing in your case it will be low. Desolder one side of both C74 and ZR7 and check again. If it's high now then either the capacitor or zener diode are shorted.
If pin 5 of U50 is still low it's time to check the path back to the PIA. The PIA pin controlling that driver circuit is pin 39 of PIA U54 (CA2) but it also goes through a buffer (pins 12 and 13 of U49.
If pin 12 of U49 is low and pin 13 is high then the 7407 at U49 is probably bad. If both pins are low then it's time to test the PIA.
As usual, confirm all this stuff with the schematics. I'm making this stuff up as I go, not just copying it from somewhere else
