How much water, just a bit, running into the drain? Get some large cart wheels (here's a picture, but get bigger than this, to keep your platform out of the water.
http://www.uline.com/Product/Detail/...B&gclsrc=aw.ds
A sheet of good plywood, 5/8" or 3/4". Cut it in half. Reinforce it with 2x4. Attach the cartwheels to the 2x4.
Put plastic bins which seal (like rubbermaid) on the frame.
Wrap plain charcoal briquets (no starter fluid) in newspapers, then wrap that in scrapbook paper (for the protection of your fabric - acid & lignen free, etc.) and put a bundle in each bin with your fabrics. Absorb odor & moisture.
... can you tell I lived with your problem once upon a time...