It's probably not the adjustment of the chain 'pusher', and in my experience the cable should still be ok at that age. If it's going to be grotty, it'll most likely be at the wheel end where the spring loaded section is. But the gear lever moves more cable than the derailleur actually uses, so the bum shifting often occurs as a result of something else.
Do check the two jockey wheels of the chain tensioner, especially the upper one. They rotate and slide on stainless steel bushings which are fairly well sealed from the elements but the clearance between the jockey wheels and the carrier is quite small and gets gunged up. Despite (and probably because of) feeding teflon lube into the narrow gap on a regular basis, the shifting on mine became absolutely appalling. I eventually had to take it all apart, washed the components, and then treated the bushings to generous amounts of Castrol Moly Grease. So far, so good!