You can wash/clean the ball as often as you feel it is necessary. It wont hurt the ball any that I know of. Difficult to give you an exact timetable because of the different types of balls/surfaces and amounts of oil or games played. I usually just watch my ball reaction and clean it when necessary.
As for the ball skidding in the 3rd game, that is probably due to carrydown. In my leagues, on my teams, all of the bowlers play different lines/angles/hook and probably because of that, oil is pushed/carried all over into my area of play and I suffer from less hook usually about the middle of the 2nd game.
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I am the SGT Schultz of bowling.
"I know nothing!"