It would sure help to know what game. Again.

Williams used at least three different styles of score reels that I know of.

OK. Later style.

It is good to make sure everything is clean and adjusted properly. 92% alcohol is a good cleaner.

I will paste this in regards to later Williams from another post-

They are pretty simple in design and function, and easy to get working if they do hang up. Only thing I have had to do out of the ordinary, is sometimes the cam that drives the switch lever develops a tiny flat spot which results in a small point on the cam. A file can be used to round it off, adjust switches if necessary, and everything will work fine again.

I use tiny dabs of super lube on the cam and the post it rotates on after I clean them, as that is where it looks like the factory lubed them, but that old lube has long since solidified.