FL-120 - Keep running after STOP button pressed.
Quick question ? - when did you last change your injection pump oil ? If it is more than 50 hours - this can be your source of the issue as the rack will stick. That said we can also look at a couple other things.
I am no Ford Lehman expert, But I do own one, but Sounds like you need to check a couple things. The rest of you FL-120 experts please chime in as to what they think this issue is.
I believe the "STOP" button your pressing sends a 12 volt signal to turn off the fuel. However you said the solenoid was moving, however it still did not stop. Even when pushed by hand. So probably not electrical issue.
The shut off solenoid operates a shut off valve or in some cases moves the fuel inj. pump control rack to the off position, others shut off fuel with it's own special fuel shut off valve.
It sounds like that valve is not closing probably due to some mechanical reason or more likely the FI rack is not moving fully to the shut off position & is sticking to such a point that the shut off solenoid can't move it. May have gone to long with out changing the INJ pump oil.
In either case, it sounds like it is an "on engine" fuel injection pump or fuel shut off valve issue.
So start there first.