Confusingly troublshot Overdrive problem

[upside-down]

I have a 92 240 auto. My overdrive dashlight will not go out in either shifter switch position. I can hear the relay engage upon depressing the button, however the arrow light stays on. I have checked under the shifter boot and the wire is not worn. However, I taped up the wire where it rubs on the tranny under the shifter. With the connector disconected from the solenoid under the car I get 12 volts in the connector when the relay is operated in both the on and off positions. When the solenoid connector is disconnected the dash light goes out. If I remove the relay behind the glovebox, while the solenoid is connected under the car, the dash light stays on. This suggests to me that the wiring is shorted to ground somewhere else. In neither situation does the 4th gear engage. Is my relay bad since I get 12v in both positions and the light won't go out? I'm suspisious about that since the light stays on when then the relay is removed. Hence I'm confused and stumped. I guess the one thing I haven't tried is to wire the solenoid directly to the battery. Also, I felt up around the solenoid, the rubber cover around where the wiring enters the body is coming apart.