I have a similar problem in my 88 All-Trac wagon. Besides the reservoir leaking, I know the motor works because the rear wiper button will produce water on the liftgate window (well, it will before the reservoir drains itself, lol). But the washer fluid button for the windshield does nothing. It does seem to draw some amperage from the battery though, as I observed my dash lights dim slightly when pushing the button and holding it while parked in the driveway with the car running.
I also installed a lights-on reminder, and when I have my headlights switched on, pushing the washer fluid button will yield a chirp from the lights-on reminder buzzer I instaleld!
Anyways, I'm getting the Toyota repair manual & electrical wiring diagram books for my wagon sometime this week. Maybe the solution for my problem will be similar to what you need to do for your FX.