I just keep a plastic container inside and drain it when the dripper is done. And yes sometimes it gets all over but i use a sponge and clean it up if it doesn't dry up, which it usually does. My husband and i started a reptile cage business awhile back and he has a design for a new cage, made of wood so it looks as nice as your cham, with a drainage system built in that hopefully will be done soon though. Right now we have the ones that are on our website for the bigger guys which makes it soooooo much easier. You just empty the hidden bucket at the end of the day..
Debby