Have we confirmed this is in fact the ball count unit ?

This stepper should use one coil to advance each "step" and the other coil will reset the unit back to "home".

If the coil is still activating - then the set of contacts that detect the unit is "home" are not gapped. so the coil is waiting for the set of contacts to be closed.

However - I'm thinking as to why this coil is activating on power on. Is the score motor running ?