Found it! But it dosen't have a rod to push. (internal)
I do not think mine has a rod to push either. You can remove the solenoid and see if it is mechanically jammed, if it is like mine.
Mine has a spring, that extends the solenoid rod when power is removed, to shut off fuel. Solenoid on mine is held in place by two screws, I think Allen head if I remember right.
I would remove the solenoid and see if the rod can be moved. It should be extended normally, but push in and then come back out on its own when you remove your finger. Yours may be stuck in and not extending to shut the engine off.
You could also just unplug the solenoid and see if the engine will start. If it starts, it is likely the solenoid is stuck in the retracted position. Unplugging would eliminate an electrical issue that was keeping the solenoid energized all the time...
IF it is the same set up as mine, of course...but it sounds like it may be.