The fans stay going when the ignition is not on also the window washer pump . The engine shuts off 15 seconds after the key is off
This would be a great one for the old radio show! Tom: Doesn’t anybody screen these calls!! … 
These all sound like a problematic body control module to me. When you press the windshield washer button, that signal goes to the BCM. The BCM then commands the windshield washer relay to do its thing. Before replacing anything though, make sure the battery and charging system are working nominally. Before first start of the day the battery should measure about 12.6 volts. Immediately after starting the engine, 13.5 to 15.5 volts. What do you measure?
Has this vehicle ever been under water? Note that on some vehicles it is normal that the radiator fans will stay on for a while after the key is turned to off. That part might be a red herring.
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