On the washing machines I have seen/worked on, the fabric softener resides in a cup-shaped object on the top of the agitator. When the machine goes into the first spin, the liquid gets thown up and outward into a secondary larger diameter cup-shoped gizmo that has a bottom inward facing lip to keep the fabric softener liquid from running down until the agitator stops spining. Then, then fabric softener liquid runs down the sides of the agitator..

That whole dispenser mechanism can be removed and placed in a bucket of water to loosen up accumulated crud, or your power washer idea should also work, just use low force so you don't destroy the plastic. There is no valve as such, just centrifugal force that transfers the liquid from the inner cup to the outer holding cup, and then lets it run down when the spinning stops.