You don't want or need the cage to stay at a constant high humidity, but create wetter and drier cycles. The cage perches and plants should dry out at least once each day to control molds anyway. What you want is to avoid severely dry conditions that dehydrate your cham. You can slow down the cage evaporation and drying by adding more live plants to hold on to more water droplets, and hanging some plastic sheeting on the back and/or sides of the screen cage.