First of all I would put your sticks in a tupperware or even a plastic ziplock bag (drop in a boveda pack or something while they are in there). I would salt test and calibrate your new hygrometer and before doing a good job seasoning your desktop box I would check all the seals and check the seal around the glass top, maybe get some food grade silicone and caulk around the inside seam where the glass meets the wood.
The next thing I would do is actual season your box (
phrasing..), put some
distilled water in a shot glass, place it in the humi and let it sit for a few days, you can always use a new clean sponge that you have dampened with
distilled water by putting that on a small plate inside your humi, close it up and let it sit, check the sponge in 3 days or so, by then your should have your hygrometer all set and you can place that in your box to check the humidity, you don't want to rush seasoning and from what I have read / heard and learned you don't really want to wipe the inside of the box down with water as it can cause swelling and slight warping that may make it tough to get a good seal.
