I hadn't noticed that you had Sram shifters and a Shimano RD, and that might be the problem, though it usually causes overshifting and poor spacing. Also, Sram makes many shifters specifically for Shimano. The rule is Sram shifters with numbers like 7.0, 9.0, or X.0, are for Sram systems, and those with names such as MRX, attack and rocket, are for Shimano.

So check that the shifter and Rd are compatible, then if it isn't a cable check both the RD and shifters for slop, and decide which, if any, to replace first. Given what this bike is, I'd go slow about sinking too much into replacing a string of components, but that's just me.