Mike, the contacts that you should be concerned with are the ones around the sides of the dial, not the ones on the back. Since each button's frequency is actually a combination of two tones, one dirty contact can distort the tone generated from several buttons. It is fairly unusual for so many of these contacts to fail at the same time, but if the phone has been stored/inactive for a long period, anything is possible.

Remember that you have a 3 x 4 matrix of tone combinations, so with only one row of buttons working, you likely have 5-6 dirty contacts. Be cautious about attempting to tune the transformers. The tones are very, very precise and when you start messing with both transformers, you may never get a perfect match again using the human ear.

Ed Vaughn, MBSWWYPBX