I typically let the prints soak in a try of water while I print. When I get a few, I rinse them in the same try with multiple changes of running water. Before hangup the last thing I do it a very deliberate wash of each print in running water right from the hose, making sure to rinse my own hands/fingers.
It could be that the spotted prints occur when something breaks down with this procedure or I get sloppy. Maybe a second washing stage tray or proper print washer is in order.