That sounds like a cable adjustment issue - the explanation of exactly what's happening requires that you know how the internals of the hub work. Do you?

The good news is that Sturmey-Archer hubs, especiallt the 3-speeds, are very easy to tear down and rebuild if it's necessary.

Edit: I somehow didn't see that you'd said the cable was slack. The issue may be more serious.

How may miles does the hub have on it, and how cold is it where you are? The traditional SA hubs are oil-lubricated, the modern ones use grease which can go stiff and gum up the mechanism in cold weather. I solved this issue by using oil in mine.