Ok. Firstly, I need to correct what I said earlier. Score reels are
reset via motor 1A AND motor 4A. 1A resets the 1000s and 10000s, and 4A
resets the 10s and 100s. Worth a look.
When you press the start button, S relay starts the motor, and because
P3E is closed, the player unit steps to the player 1 reset position.
Then the P3A/B/C/D close and P3E opens. Theoretically, the player unit
cannot advance to the player 2 reset position (one more step) until all
player 1 score reels zero, or P3E closes. Manually advance the player
unit until P3 is riding on the cam. There should a fair sized gap at
P3E. If the gap is too small, or the solder tabs are bent and almost
touching, there could be an arc which steps the player unit prematurely.
Check that the strips of insulating paper between the switches are
intact.
Still no joy? Have a look at the switches at motor 1A. The inside switch
resets the score reels and the second switch advances the player unit.
If these two are shorted the player unit could step up before reset is
complete. Same deal with the solder tabs and the insulating paper.
Coincidentally, this could be the cause of your second problem, causing
the player unit to advance when the score motor runs.
As always, tighten the switch stack screws before touching anything
else.