I am challenged concerning the magical flow of electrons but before installing the bypass plate I would do a couple of things...

All of the relays in our cars are questionable. The OD relay is tucked just to the left of the glovebox opening and is visible once you remove the glovebox itself. Simply remove all the screws surrounding the edge; visible when the compartment door is open. If you know how to solder remove the relay cover and look for a bad solder joint. Simply reflow any suspect joints or just do them all. The majority of OD problems can be traced to the relay.

With the key on the relay is acivated by a momentary application of 12v. The switch on the shifter does that. Much like a doorbell switch. Your problem does not sound like a switch problem, but a test light probe of the wires should find one that comes on when the switch is held in.

The relay supplies 12v to the solenoid. From under the car you will see the wire that runs to the solenoid. Probe that wire with a test light connected to ground. The overdrive button on the shifter should act as a switch for the test light. Press once and the light should come on, press again and it should go off. If it is supplying the 12v on demand the solenoid should be clicking in tandem with the test light, easy to hear when you are laying beside it.

If the solenoid is clicking in response to the application of 12v from the shifter switch and the OD is not engaging then it is probably time to install the bypass plate.

If you install the plate be sure to clean up the area before removing the solenoid as there is an accumulation of dirt around it after almost twenty years of riding under the car.

I am still betting on the relay being defective.

Good luck.

Randy