I seriously doubt it's worn out, I've done >36,000 miles on a similar set and they're just fine. If the shifting is ****ty, there's a good chance that the cable inside the shifter is frayed and binding on the inside of the shifter. Mine usually last around 3000 miles before heading south and as soon as it starts being hard to shift or hanging up on downshifts (clicking the small lever behind the brake lever) I swap out the cable. If you don't then at some point it'll break and that's no fun 25 miles from home.
First, buy a set of Shimano inner and outer gear cables from Amazon or likewise. You can get cheaper, but Shimano cables are really nice.
Then put the bike into the lowest gear at the back. Undo the bolt holding the cable into the rear mech. Does this have internal cable routing - if so stop, let me know and I'll tell you a good way to do that.
Pull the hood on the shifter forwards towards the brake lever. Underneath on the outside of the shifter, you should see a small opening where the cable goes. On the inside you should see a small plate with a screw on the opposite side to that opening. Take it off and don't lose it like I did.
With the cable tension released at the other end, you should be able to push the head of the cable out of the shifter. Note you can only do this with the shifter in the smallest gear at the back. Don't shift the gears until you're finished.
Remove the old cable and install the old cable. Thread it through. Don't add any oil down the outers. I'd recommend replacing them (it makes for lovely shifting) but you probably won't have to.
When you have the cable going through the rear mech, push the mech ever so slightly into the wheel and snug up the bolt that holds the cable.
With the bike on a stand or someone holding it up for you, click the shifter into the next gear, turn the cranks and unscrew the adjuster on the mech until the chain climbs into gear. Shift back down and try again. Then try shifting two gears, unscrew the adjuster until it will shift both gears in one go. It'll take a bit of fiddling but it's pretty straightforward. After a few rides you'll need to repeat this as the cable stretches.
Cut the cable to leave around 2-3" sticking out and cap it with one of those little alu caps.
Should cost you around $25-30
To add to ddickey above DO NOT disassemble the shifter. It's full of very small very fiddly pieces and springs that will disappear in the blink of an eye. Shimano shifters literally last forever, I have some on my mountain bike that are old enough that I can't actually remember when I bought them (15 years ago? 20?) and they're still working just fine.
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