Don't have a schematic in front of me, but most electronic pins had the same "premise" of a coil being driven by a logic gate, predriver transistor, and darlington transistor.
Working backwards, a coil that fires on power up USUALLY is a shorted driver transistor on the main board. Again, I don't have a schematic in front of me, nor am intimately familiar with DE pins, but I suggest you might look at all your output transistors on your MPU board with your meter set on DIODE.
These transistors are usually TO-220 packages (big transistors, three legs, standing up, metal heatsinks with a hole in them at the top). Do diode tests between the pins in every possible combination - you should either get nothing, a reading between 0.5-0.7 volts, or a "0" (short) - THAT'S the bad one, you should not have any of these shorted.
These DO burn up also, when they have a lot of current go through them for an extended period. Visually check all the transistors, if it looks burned, chances are it has failed. Good luck.
Also, a hint - power up your game with the connectors near these transistors DISCONNECTED. The game should boot without any solenoids engaging. Now, plug in connectors ONE at a time, and find the one that causes the "instant-on". Trace back from the connector to the group of transistors that feeds it, you shoudl find your bad one.
Chris