It is a sign of old rubber and I believe what occurs over time is the as the rubber ages, it breaks down and the sticky feeling is one of the compounds that comes to the surface. It's a catch 22 situation since as you clean off the sticky layer, you expose the rest of the material and it tend to dry out quickly, yellow and crack, but if you leave it alone, then the sticky smearing just tends to get worse.
Cleaning it with Acetone will work, or rubbing alcohol, then you might try a rubber conditioning solution like the stuff in auto parts stores. Best solution is to have the rubber parts replaced, but as you pointed out due to age, it may not be possible.
Paul Caldwell