Since the dash light comes on, the relay and the wiring to the shifter are just fine. The dash light on your vehicle is connected after the relay parallel with the solenoid so if the dash light works, the relay is reacting properly.

And since you have checked all the connections to the solenoid, and the solenoid clicks when the shifter button is pushed, you have only two possible causes for your problem.

Low/old tranny fluid. Low/old fluid loves to get whipped into foam and an air/oil mix just doesn't work well in the OD.

The other item which can show up after about 130,000 miles is worn actuating piston seals. OD has to come of tranny, but tranny can stay in vehicle. Go to the Features at the top of a Brickboard page, click on 700/900 FAQ, scroll down to M46 tranny section and click opn the OD section. Tells you what is needed to do it yourself.

Duane