Speaking generally, once a coil has been energized, and it needs to stay on for a while (for scoring purposes, usually), the coil will have a lock-in circuit that keeps it energized. The lock-in circuit will have, at minimum, a switch on the coil itself and a score motor switch that will open somewhere in the score motor cycle. Sometimes instead of a score motor switch, there is a switch on a stepper unit that opens when the stepper is at one end of its movement). It sounds like maybe that score motor or stepper switch isn't opening, which is keeping the coil energized.
I wasn't sure from your description whether you meant the 2 kickout holes that shoot the ball up to the moving target, or the drain hole at the very bottom. But there are several examples of that lock-in behavior in the snippet below:
SM eject coils (resized).JPG
If you look at what happens to turn on and keep on the Shooter relay, for example. The ball must close one of the 2 the shooter switches when it falls into either kickout hole. When one of the shooter switches closes, it completes a circuit to the Shooter Relay, turning it on. The shooter relay has several switches on it labeled "Shooter Re." that are now opposite the state drawn on the schematic.
One is on the Shooter Relay itself and closes, keeping the relay energized because it's in series with a n/c Motor 5B switch (left side, mid height on the snippet). That's the lock-in circuit for that relay.
Another switch on the Shooter Relay causes the score motor to turn:
SM score motor (resized).JPG
Another switch on the Shooter Relay closes in the Shooter Coils circuit and turns on the Shooter Coils briefly when the Motor 4C switch temporarily closes. Motor 4C closes before Motor 5B, so after the Shooter Coils fire, the Motor 5B switch opens and relaxes the Shooter Relay.
In this example, if the Motor 5B switch didn't open, the Shooter Relay would just stay powered on, the score motor would continue to cycle, and the Shooter Coils would fire and keep trying to eject the ball every score motor cycle.
Nice redrawn schematic on the ipdb site! Wonder who did that?