I had to move my stash from California to Florida. I still have the Rubbermaid totes, just in case. You never know if there will be a roof leak or something.
All the fabric is now on shelves where it gets air, but when it is boxed up, I keep the silica gel packs in with the stash so it can't get moldy. The gel packs are the ones that come in new boxes of shoes.
You might want to do this if your stash is sealed in plastic.