Apply Fray Check, or similar product, to the fabric edges. I frequently use it when I want to reinforce a clipped spot.i don't remember if it shows when dry or not. Test on a scrap to see how it does.
Fray Check works well, but leaves a hard edge when dry. Applying it to every edge will probably be tedious, but perhaps easier than replacing the pieces that are fraying.
There are several brands available. One that comes in a tube leaves the fabric feeling softer after it has dried.