Yes even "I" have to deal with the issue of rot from time to time, I know many of you throw these pieces of fabric away, why I don't know but I have heard from many of you that you do so. Me, well I'm weird in that I look forward to the holes, the tid bits scraps left in the washer and dryer, as they are layers in my mind.
But what to do with the larger pieces that are thin and fragile? Simple back it with a sheer iron on interfacing and go forward! If need be used double sided interfacing and press it to a stronger piece of cotton fabric or canvas, when I've needed to do this I've used cotton flannel fabric as I love the way it needles for hand stitching.
Some tidbits of fabric from the dryer, I have a pile I collected from the washer drying on a rack.
More holes and I can't wait to stitch them!
Later this evening this batch of fabric will be placed into a dyebath of cutch and walnut, once it's achieved the appropraite tone of "sandstone" I will wash it, dry it, press it, and then cut it down to size. After all the logistics of placement is figured out I may back it with a sheer interfacing such as Misty Fuse (my favorite) as I really don't like putting netting/tull over the face of my textiles.
If you want to throw your fabric away I understand, I really do! It simply does not fit with your vision and I know how difficult it can be to work with fabric, fibers, yarns, threads, etc., that do NOT fit in with your vision. If it's way cool rotted stuff just send it to me instead of throwing it away, I promise I'll turn it into something wonderful!