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As your washer works I am assuming the wiper motor is working as it drives the mechanical style washer pump, that and the fact that the wipers move up and down an inch or so.
I would inspect the linkage from the wiper motor to the wiper arms. I suspect that the nut that secures the linkage to the motor is loose, or the linkage itself is damaged at the conection point.
That is assuming that both wipers move the same amount and together.