How humid is it in the cage? How often are you misting and for how long?
Yes you do want the branches to dry out. Otherwise you could get mold, rot, fungus (on the branches and on the chameleon)
If you dont have a drainage system, a simple pan to collect/catch the dripper water will work (empty pan at least daily, more often if needed). A screen overtop of the pan may be advisable if you have a small chameleon or free-roaming insects.
My cages definately do dry out completely between mistings, since I only mist once or twice a day. The drippers provide most if not all of the drinking water. I use aquarium air hose to direct the dripping water. The chameleons drink directly from the dripping hose. The hose ensures the unused drips fall in a controlled manner, to the location of my choice (the collection pan). There's usually a little bit of splash even so - especially if a chameleon decides to stand under the dripper, causing the water to fall all over the show. I use towels (changes out frequently) on the bottom of the cage to capture what doesnt hit the collection pan.
