I have an AP cage and keep my humidity between 55-65, closer to 70 when in shed. I also have two little plastic dishes and keep moist sphagnum moss on each side of the cage (the dishes keep it from mildewing the mulch) when needed.
If he dumps his water or right after I refresh the moss the humidity will jump to the 80 range, but falls over a few hours. This time of year I don't have much trouble since the ambient humidity is so high. His cage right now is reading 61% and 75%, and the room ambient is sitting at 67% - his moss is dry. The higher side just happens to be closer to the water bowl and the lower side has a CHE running.