If you cannot pull the slack out at the derailleur, there might be rust binding it in the rear housing loop (chainstay to RD). Disconnect the cable from the derailleur, and pull it away from the frame to pull it free of the rear housing. (you'll have to cut off the crimp if there is one) and then see if you can work the shifters against tension you apply by hand to see if the shifters work and the cable moves freely and smoothly in steps as you work the lever.

If the front end and shifters work OK, you might be able to keep the cables, only needing to oil the rear loop, but most likely you'll gt the best performance by replacing both cables completely, including the housings. I emphasize the housings, because in most cases the inner wires will outlast housings, who's liners wear through. Good quality gear cables with stainless die drawn wires, and lined housings aren't that expensive and worth the small difference above the less expensive stuff.