Brian,
A quick check to see if you have a electrical problem with the solenoid, 3/4 switch, ground connection, and the interconnecting wiring;
Disconnect the wire from the overdrive relay that goes to the 3/4 switch then on to the solenoid. I don't have a wiring diagram handy but you should be able to find it.
Attach a lead of your ohm meter to that wire.
Attach the other lead to engine ground.
You should see around 6 ohms or less when your shift level is in 3rd or 4th gear.
If you see more than 12.5 ohms, then either the pick coil of the solenoid is not in the circuit ( open coil or the internal switch contacts in the solenoid are not making ) or the 3/4 switch contacts are getting so bad that it is only working marginally. In any case if you measure more than 12.5 ohms from that disconnected wire to engine ground then you will have to ohm out everything in the path. As Bill said, The solenoid should be around .5 ohms if you check it alone out of the circuit. If it is above 12.5 ohms then either the pick coil has burnt open or the internal switch contact is not making for
some reason. If the solenoid is above 12.5 ohms while still installed on the o/drive but if removed shows the correct .5 ohms then something is preventing the solenoid plunger from moving all the way down to the bottom rest.
One other quick check is to put one lead of your ohm meter on the little ground wire to the RELAY on the firewall. Put the other lead on the engine( a nut without any paint or rust). If you don't see close to zero ohms, then you have a poor ground connection from your engine to your overdrive relay. On a BJ8, it is a screw on the firewall that has several black wires under it, with a star washer between the bottom wire and the firewall. The paint should be scraped off of the firewall under that washer, ( not something folks want to do after painting the car to perfection but necessary for a good ground )!
Lets us know what you find .
Ed