When the cable starts to die, it will more likely not really stay in any cog easily or it will tend to LOSE tension, meaning that it will want to stay in the smallest rear cogs. But even if the cable is still ok, the housing might not be.

It sounds to me more like the shifter is starting to go and needs to be flushed (high pressure water and WD-40 can work, but relube after). Does it feel different to shift?