I have a '95 E150 which is probably similar to yours. The Windshield
Wiper Motor is operated through the Wiper Control Module. You are
probably hearing the WCM, which is under the dash on the right hand
side. It has a 14 pin connector. I cannot tell from the simplistic
schematic whether the WCM's innards are solid state or
electromechanical. Since you are hearing a "click" I suspect there is a
relay not doing its job because of dirty contacts, or a broken wire, or
bad connector mating.
Because you are hearing a "click" the signals from the column mounted
Multi-Function Switch are probably okay, but do not rule it out.
What you didn't want to learn: besides the connectors at the
Mulit-Function Switch and the WCM, there are three (3) connectors in
between. There is also a connector between the WCM and the Wiper Motor.
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