I would not use plastic totes, as you stated humidity can cause mildew quite quickly. If using cartons get acid free ones. Place the fabric in pillow cases or some sort of fabric bags. One option I just thought of would be to put a number of those little silica packets ( like you find in shoe boxes,etc. to absorb moisture) you could put a couple of those in to fabric bag/pillow case with your fabrics then place into those vacuum bags ( space bags) remove the air, then place those into your totes. I would toss in a few more moisture absorbing packets just for added safety. The space bags will help you get more into each tote. I've stored items for over a year in them before.