So it sounds as if the the 0 - 9 coil is remaining energized when a run is scored. I went through a few
Williams Pitch & Bats, and the 0 - 9 drum unit energizes from the Run Relay. The Run Relay should be
in the backdoor. When this happens, you'll want to check if the Run Relay is remaining energized..
I'm thinking that it probably is, and if so, the Run Relay is energized from the Run Switch which (if
I remember correctly), is on the Man-Run-Unit near home base. Sounds like the Run Switch is maybe
sticking, but if it is staying stuck, it seems the Run Relay would energize immediately at first start up, but it"s
not until a run is scored, right?

This is what has me wondering a little bit. You might want to make sure that the contacts in the Game Over Relay are
adjusted properly, and also, there's a make/break set of contacts in the 3rd Out Relay you'll want to check for true
adjustment. There may be other issues, but resolving this one first may be the best bet..