Sounds like you may have a score reel or 2 (or more) that while showing zero in the display is not actually at the zero position with regards to the switch on the reel. I would remove each reel and actuate by hand to make sure the switch is making and breaking when it should. If it appears to be then it may just be a very dirty switch that needs a flex stone cleaning. Grab a score reel that is already working correctly and watch how it actuates the switch on the score reel when in position 1-8, 9, and zero.

I will add Chicago Coin score reels are notoriously finicky...I have had 2 Hollywoods.

I'm guessing this machine has the board instead of switches.These can be a pain. If you go to the back during reset you may be able to find the culprit by nudging the score reels.If not,I would suggest going through all of them. I had to on a rawhide.Take the board assembly off the reel and cleaan the board and ccontacts .The make sure it's put together right.It will have a small notch where the contacts go on the reel.Also check your connectors.Might want to clean them as well.
These machines have been around for along time,so they need going through at some point to get them working properly.