I suspect you have a couple different things going on as stated.
The GEM produces a reference voltage. If the wiring is good, and door properly latched shut, switch working properly, a closed door closes the circuit, completes the circuit to ground through the door latch, so there is only a single wire to it needed to do that. The GEM senses that its reference voltage was pulled down with the door properly shut and all related parts functioning.
You stated you found the wire shorting out but that would only (momentarily, not dependably) cause the GEM to think the door is shut, while you have the opposite situation where the GEM seems to think the door is open when it isn't, so you have either a latching issue or the switch itself is failing, or the wire is frayed if not broken, so the switch is intermittently (if ever) failing to short the wire to ground when the door is shut, causing the bell and lights to come on.
See attached wiring diagram. A multimeter would be handy to check continuity between the switch terminal and ground in the latch shut state, and test continuity between the wire to the switch terminal, and the GEM to see if it might be frayed apart in the door hinge, boot area which is another common fault.
Another possibility is the door is sprung or misaligned and the latch is not fully closing so it is not fully depressing the switch to close the circuit to ground. I suppose if your door latch striker sleeve is shot, or the latch mechanism itself is falling apart (latch itself is probably the least likely of all alternatives presented) it might also cause some play and the same result.