You may have one or more problems occurring if the slippage is in more than just high gear. The high gear should work properly with the cable totally disengaged. The fact that the indicator moves means that the clutch that engages the different parts of the hub to produce the different gears is moving outward, probably due to wear on the pinion pins or the clutch, or due to a weak/broken spring. In high gear the clutch is pushed down by the spring so that is sits on the surface of the planet cage and the clutch pushes against the ends of the pinion pins. If the pins ends are badly rounded off or the clutch is damaged you can get slippage, especially with the higher pressure in 3rd gear. But the pinion pins are not a factor in other gears and one is not depending on the clutch in the same way in other gears. Because of that I think the most likely cause is the spring, which if weak or broken (or if the cap is missing) could cause insufficient pressure in 3rd gear and misalignment in other gears.

I would replace the spring no matter what, as you can't tell for sure from inspection if it is weak. In fact if you happen to have a source nearby it can be replaced by merely removing the right cone, without the need to disassemble the entire hub, so that you can see if that cures the problem.