I have a thought on the suggestion of leaving the binding on and butting up the new backing to it. Wouldn't the original binding be at risk of coming loose if it's attached to the old backing which is shredding?
I'd put a new backing over the old. I'd take out the handstitching on the binding, open up the binding and sew over the original machine stitching that attached the binding in the first place, this time catching the new backing. Then hand sew the binding to the new backing and tack stitch with my machine to hold the backing in place.
This has been suggested previously, but I thought I'd add the part about the original binding coming loose. Whatever you do, good luck.