That's a reliable test, and it sounds like the solenoid is bad. It should make a very definite click -- there's a high-current coil that makes it snap home, and then a low-current coil keeps it there. If it doesn't snap all the way, the contacts on the high-current circuit don't open and things get hot fast.

If you remove the rubber boot, there's a plastic cap under it that comes off by undoing two small nuts. Have a look at the soldered connections in there.

For the benefit of others, you can do a quick check that at least partly isolates electrical problems:

1) Turn on ignition, but don't start the engine.
2) Toggle overdrive on -- you should hear the relay under the hood click.
3) Ease shifter into 4th gear -- you should hear the solenoid snap over.