It sounds like your player unit is having some trouble either stepping cleanly, or some of the switches in it aren't changing states as they should. If you remove the screw in the upper right corner that holds the player unit in place, it will drop down 90 degrees or so. From that position, you should be able to see all the switches clearly. Take a good light and get in there carefully and clean the contacts. After you've cleaned them, make the unit step around and watch each of the switches to see that it's opening and closing as it should. When they're all opening and closing correctly, look at the back side of the switches and make sure there aren't any tabs or wires touching or disconnected. It's really important to be careful and thorough when doing this.
When you're done with that, screw the player unit back into it's normal upright position and work the plunger with your finger to watch it step around. Does it move quickly to each new position? Are the rivets and snowshoe contacts clean?
I realize this is kind of a holistic way to approach your problem, but if you can do all that carefully, it may fix your problem and help you avoid some future problems in that area. If it doesn't fix things, at least you know the switches and stepping is working as it should, and you can look elsewhere. But it does sound like your player unit could be the problem.
Make sure those player 1 score reel switches are making good contact as well, opening and closing correctly.