I don't do anything with them. I put them in my pocket during season and when out of season I keep them in in a plastic container or a little nylon pouch. I use them hard. If they break or loose pitch then they go in the trash and I make or buy new.
It seems that predator hunters are their own worst enemies by over thinking everything. Practice being a good woodsman, scout and study the habits of your prey if you want the best results. You don't have to have a specific rifle, call, camo and so on. I have won state calling competitions and finished 2nd and 3rd in the world for calling. That doesn't mean crap except I know sounds and calls. Some of the best hunters I know sound like crap and use low end equipment, chew tobacco, smoke or don't even wear camo. Don't take this the wrong way but there are more important things to being a successful predator hunter than how to store your calls. Having said that, if you find it a valuable tool than do what works best for you.