Also the switch on the gearlever, or the connections to it, or the wires at the bottom where they flex as you change gear.
The simple starting point is just to wire a temporary power lead direct to the solenoid from the battery -does it click reliably on/off?
Then try the same thing, but with a temporary switch inside. Try driving around and operating the O/D. If everything works properly then, you know the fault must be exyernal to the unit itself.
DON'T leave the O/D engaged in lower gear, and especially NEVER in reverse.

Also try changing the gearbox oil, clean out the overdrive filters, and refil overfull by tilting the car.