The govenor seldom gives problems, so I'll ignore it at this time. If you want a trouble shooting guide on it or the kick down switch, let me know. The solenoid and relay are the ones that usually give trouble. The following tests are all made with the engine not running.
To check the solenoid circuit:
Step 1. Ignition switch on. Ground the TH-SW terminal on the relay (it's usually the terminal on the right as you look at the relay). If both the relay and solenoid click when you make and break the ground, the circuit is working properly.
If it doesn't, go to step 2.
Step 2. Check the fuse. Just because it looks good doesn't mean it is. Many of them break in the metallic end cap and still appear good. Use a continuity light or Multi-meter. If in doubt replace it. Repeat step 1.
If it still doesn't test good, go to step 3.
Step 3. Ignition switch on. Check the IGN terminal on the relay with either a test lamp or a multi-meter. It should be hot. If it isn't, the trouble (or at least one) is between the ignition switch and the IGN terminal on the relay.
If it is hot, go to step 4.
Step 4. Ignition switch still on. Connect a test lamp of the circuit voltage between the SOL terminal on the relay (it's usually on the left as you look at the solenoid) and a ground. The light should light. If it doesn't, the relay is bad.
If it does light, go to step 5.
Step 5. If the relay is working in step 4, the solenoid should snap when the TH-SW terminal is grounded (see step 1). If it doesn't, connect a jumper from the SOL relay terminal to the short black wire (it's only a couple of inches long) coming from the relay (some may be blue but all have an electrical bullet type connectors on them). If the solenoid does not click when you connect the jumper, replace the solenoid. If it does click, the wire between the relay and solenoid is defective. This should pin point the trouble.
Just remember what voltages you're working with. Lots of guys have converted their flatheads to 12V and are running a resistor for the 6V overdrive. You'll smoke the relay and solenoid in short order if you inadvertently give it 12V. Relays from old 6V Studebakers, Ramblers, Nashs, Hudsons, Chryslers, etc. will work. Some of the Nashs have 6 wires. Just mess with them and you can figure out which ones are needed for your Ford. Solenoids are something entirely different. I've used the shafts from a bad Ford solenoid in a Stude or Nash solenoid and got them to work. Just pay attention to which way the flat sides of the shaft lay on the Ford and reassembly the same way. Studes and Nashs are fairly plentiful. I sometimes find a complete trans for $5-$10 at flea markets. I just strip the O'D pieces off and trash the rest. Pieces I save? The planetary gears, sun gear, free wheel unit and its bearings, the govenor, and solenoid are all the same in these BW overdrives. Virtually all the other pieces won't work in a Ford O'D. Auto Zone can get you new solenoids.... $120 or so! So the solenoid from an old Stude is easily worth $5. Also watch the mating surface of the solenoid. Some are flat and there are a few with an 1/8 inch step. You can use a 12V solenoid and relay and do away with the 6 volt stuff. The govenor is not voltage sensitive... it's basically just a switch.
Hope this helps some.... rumble seat