Hi Zeegske
in Post-1 You write "... not really into EM machines ..." - the EM-Folks use some "wording, terminology" and Your text (well, to me) is a bit confusing - I am not sure "do I understand what You want to write" - Do You strictly talk about starting a new game - and while the Score-Drums do step forward towards Position-Zero - at starting a new game the fault shows up (Score-Drum in the Backbox, Player-1, 10'000, 20'000 etc.). When You write "the score goes from 9000 to 10'000" I am tempted to think "You are playing ball-1, making points". EM-Folks do talk about "a Score-Drum does reset - or (when there is a fault) the Score-Drum does not fully reset - in reality in all EM-Pins the socalled "Reset" actually is a "stepping forward (1,2,3,4,5,6,7,8,9) and then to position showing '0'.
My Interpretation of Your description is: At startup when the Score-Drum-10'000-Player-1 happens to be in Position-Zero, '0' --- all good. At startup when the Score-Drum-10'000-Player-1 happens to be in (example) Position-Four, '4' --- the Score-Drum is NOT stepped - the Coil on the Score-Drum is constantly pulling.
IF (if, if) my Interpretation is "right": Look here https://www.ipdb.org/files/245/Williams_1977_Big_Deal_Instruction_Manual.pdf on page-13 (Coil Chart) - M29-900 No. 3 Score-Reset-Relay Insert - "Insert" means Backbox, "Mech. Panel means bottom of Cabinet". Then look further down in the ipdb-manual - on page-32 (original-numbered-30) You see a drawing of the No. 3 Score-Reset-Relay --- there in the drawing "Switch 2A with wire-Y-Yellow, BR-BLU-Brown-Blue. So check in the Backbox in Your pin on the No. 3 Score-Reset-Relay the Bottom-Switches. Have the pin toggled-off and the main-power-cord unplugged (Safety Reasons) - press and let go the armature (You simulate relay pulls, let go) --- look at the switch - faulty always closed ? Greetings Rolf