Since you mentioned continuing on to the next player, I am assuming you have the Big Deal version made in 1977 and not the 1-player model from 1963.
Williams EM drop targets of the late 70's era had a small circuit board behind the bottom of the drop target with a copper trace that had 3 distinct sections. Someone correct me if I'm wrong, but I seem to recall the scoring should only occur when the copper arm with the contact face was sliding past the middle section of copper trace.
Is the drop target possibly hanging up and not going all the way down (where the contact would remain clear the middle section) ? Is it possible there is some conductive grease on the circuit board that is effectively shorting the lower and middle copper traces as one?