I'm not certain how young your young cham is, but the sooner you move her from a 10-gallon tank, the better.
Since you don't actually see her drinking, you probably should monitor her urates and eyes closely. Nice bulgy eyes and snow white urates are signs your cham is properly hydrated.
Misting is good for maintaining humidity levels, but chams need way more than misting to stimulate their drinking response.
I suspect she will avoid ice cold water dripping from the top of the tank. Remember, in the wild the water they lap from leaves is warm not ice cold. You can devise a suitable dripper from just a dixie-type cup with a pin-hole at the bottom, filled with warm distilled water, until you can provide a more suitable dripper system. Since she's in a tank, make certain you have a bowl or something at the bottom to collect water and don't let the water sit.
If you must rely on misting for drinking, keep misting with very warm distilled water until you actually SEE her drinking. I often think "misting" is misleading. My chams respond to "raining', rather than just "misting". 