It is my understanding that our local branch of the Humane Society does not want the scrap beds. Other shelters may have other feelings but I would check first with the one(s) you are interested in.
I've tried various storage methods and various projects to deal with tiny bits of fabric. I started out with a hamper that turned into a terrible snarled mess that I didn't want to deal with. I tried cutting down into usable widths of various sizes. After 10 or so years of that, a couple of years ago I decided that I simply do not want to store or deal with pieces under 6.5" wide because I can always cut down, but I can't cut up. It's a size usable for almost anything, is easy to store and sort.
I think the methods for cutting pieces down into like sized bits works a lot better if you have a consistent fabric style, like my friend who pretty much always works in batiks, that means her scraps are batik and they go together well. Or if you like traditional calicos or civil war or juvenile or 30s or whatever floats your boat. But my fabric tastes go all over the place and although I can put everything in one top, it's a lot harder when "everything" really is everything...
I've found a crumb quilter on the boards and she now gets use of my string/straighten the grain cuts, my fussy cut scraps, and anything smaller than that 6.5" strip I keep. A lot of what I send isn't really crumbs, but again, she can cut down and I can't cut up. I keep a flat rate box next to my cutting area and fill it up with strata of my cutting. It's like an archaeology dig going through the layers in the box and looking at the projects I made... Some years I'm more productive or clearing out more stuff than others, so now I'm pretty much out of all my little strips and my boxes are fewer, maybe one a year now instead of every couple of months when I was actively clearing out all those strips I had...
Yes, I pay for postage -- as I say it is the price I pay to set my fabric free and to encourage me to think before I spend. I have no guarantees if anything I send gets used, but I know some of it does because I've seen projects. But it was fabric I no longer wanted to deal with even if it was usable. Sure, it happens that as soon as I mail someone a box of something a perfect use comes up with what I just sent out. That's going to happen And it's going to happen that some of what I send is ultimately not wanted. At least we gave it a chance!