There are a few options. If you want to try it, the sticking problem could be the cable gummed up in the housing or even the trigger gummed up. I have loosened the cables, pulled them out and shot WD-40 though the housings and that sometimes helps. I have also taken the dust cover off the shifters and used WD-40 and then Tri-flow after things have dried out and had good success. But if the shifter is worn out, you need to replace it.
Is it 8 speed?
John