This happens not just with the limbsaver pads, but with a whole host of synthetic / rubber based materials made in China, usually when they're about 5 or 6 years old. YMMV. I've had this happen with everything from hard plastic knife handles, to plastic Bluetooth speakers, "rubber" recoil pads and grips (not Pachmayer or Hogue, and my one Kick-Eez is still good), to tool handles. Has nothing to do with where the item was stored.
They can send you a new one, but a few years from now you'll be in the same boat.
This really works, and seems to prevent it from happening again: get some denatured alcohol, but even plain rubbing alcohol will work. Use a t-shirt or other good finished cotton/linen shirt or rag that won't leave lint, because of the sticky rubber. A square piece of high-thread count cotton sheet is perfect (just cut it from the bottom of the bed, your wife will never miss it! ). Painters rags used for applying stain are already the right size and texture.
Don't use anything like a red shop towel; you want something tightly woven and "finished" that will hold up to hard scrubbing on a rubbery, sticky surface without leaving any of itself behind.
Soak the cloth in alcohol and start rubbing the item, working small sections at a time, keeping the cloth liberally wet with alcohol. If you try to do the whole item at once it makes a mess, and makes it harder to know when you're done.
Periodically dry your hands on a separate towel, wait a minute or two for any alcohol to evaporate off the surface, and recheck the small sections you've done for tackiness. You'll know when you're done.
Let us know how this works for you. I think you'll salvage it fine using this method.