I agree with not covering up the odor. Is your room by any chance in the basement? Moisture of any kind generates odors and is also bad for fabric.
Charcoal is really good at absorbing odors. Dr. Scholl's Odor-Eaters shoe inserts with charcoal are a really easy way to use charcoal inside plastic bins. My vintage featherweight case had an odor that would *not* go away, no matter what I did, until I added one of those shoe inserts to the box. It completely cleared up the odor.
Another thing you can do is add some dessicant to each box -- like the little packets that come inside vitamin pill bottles. You can buy it in bulk at a place like Walmart, as it is used to dry flowers (you bury the flower in dessicant and the dessicant pulls all the water out). I haven't used this myself but, if it's sand-like rather than flour-like in texture, it should be easy to whip up some cloth packets of dessicant for this use.