It sounds as if the vulcanisation of the rubber may have deteriorated.
Here is more than you probably would ever want to know about vulcanisation . The breakdown of the sulphur bonds can be caused by bacteria, according to Wikipedia.
BTW, I once renovated an old reel-to-reel tape machine where the drive-belts could be rolled-up between the fingers into sticky round blobs!
None of this bodes well for a solution. I would take the advice of those who suggest replacement.