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Does it spray washer fluid (at least until the reservoir runs dry) while the wipers run?
The "fluid control" button at the end of my stalk flew off which caused the wipers to go on, and spray the windshield. Check the end of the stalk if that is the case. I was lucky, and able to find the button and springs sitting in the passenger seat, and door seal.
If the wipers are running without fluid spraying, then it could be that your wiper motor is wet from the drain "udders" being clogged.