Gear shift rubber cover liquified, post cure possible?
Once the rubber has gone gummy, its game-over for that part.  You have two real options.


Cheap answer - Cover the gummy partI've put cheap heatshrink tube over some bar grips that were deteriorating.  That was ~5 years ago and they're still working fine on my MTB.   Since its not a round part, you might get away with a layer of tape.Downside is that the goop will soak through most tapes, and plastic tapes will be harsh on your hands.
Outright replacement of the hoods. Find your shifter's model number and start searching.  I see plenty on ebay, selling for 10 pounds to 20 dollars.  Expect to pay more for brand-originals, but they're more likely to fit.




Doing nothing is not really an option - they'll only ever get worse, sticking to you and your skin and gloves.

You might choose to cut the old hoods off, and ride bare, but that will expose some of the internal gubbins to the elements and to your sweat.  Also this would give sharpish edges that will chafe and irritate your hands.

I'd suggest you tape them temporarily, and start searching for new replacement hood covers.