You're on the right track in starting with cleaning the switches. A product called Caig Deoxit works well without removing the switch. It's available at many electronic supply places but it's not cheap for the little container you get. Pull the knobs straight out (very difficult in my cars) then spray into the switch innards.
You can use it on window switches as they start to get flaky.
You'd think I was a Caig spokesperson by the number of times I've suggested Deoxit over the past couple of days.