Sounds like your stash is just like mine was until about a month ago when I ran across a site that gave suggestions about how to cut stash into usable-width strips for storage. Since your pieces are on the smaller size (like mine), the wrapping- around- the cardboard thing probably isn't what you need. Here's her advice:
Cut strips in the following sizes: 1.5", 2", 2.5", 3.5" (or whatever width you seem to use the most).
From 1/2 yd of fabric, you can cut the following: three 1.5" strips, three 2" strips, and three 2.5" strips.
Anything 6" or wider, store with fat quarters.
Then, when you are ready for a scrappy project, all those strips are ready to be cut into the size squares (or whatever) that you need.
Of course, this takes TIME, but I accomplished a little each day, and I'm retired, which certainly helps! I then put my strips into labeled zip-lock bags by color and stored these bags in some of those cheap plastic stacking drawer units that you can get at WalMart, Target, etc. I am so HAPPY when I walk by my scrap stash now, and I'm motivated to get busy on some super-scrappy quilts, which I love.