For what it's worth, I have had the same symptom many times with my Healey and seen it on others: I believe it is usually caused by those tiny little contacts in the solenoid that select the "pull-in" windings on the solenoid. When the contact resisistance on these is more than a mere ohm or so, the pull-in current is way too low and the solenoid just cannot generate the force needed to lift that little ball in the hydraulic system and engage the OD. Testing the solenoid statically in the car and seeing that it moves the lever does not prove that it works: the force needed to move that lever is much higher when the car is underway and there is pressure in the OD hydraulic system. If you want to test it in your shop, jack up the rear wheels, put the car in high and bring the car up to speed to see if the solenoid can move the lever and engage the solenoid. You will know you probably have this problem if the OD will engage if you manually move the lever, but won't do the same with the solenoid.
I doubt if adjusting the linkage will make much difference in your car -- it worked before and there is little reason for those adjustments to have changed. I would sure check it out, but would not expect the problem to be there.
This electrical contact problem is difficult to diagnose, and if you work the solenoid by hand, it can change the resistance of those contacts, making your overdrive work or not work intermittantly. I have had this problem on the road and it can go away as quickly as it started, and the OD starts working again, sometimes for days at a time. This may happen to you!
If you have this problem, the only sure cure is a new solenoid (these are costly). I have filed the contacts and had limited success with that, but you never know if that will last.
Also make sure the ground connection on the solenoid is good, all that current needs too pass through the ground and the bottom of that solenoid is usually oily and greasy -- this is a Healey after all....
Good Luck
Bill S