I am just facing a similar problem in a quilt I made for my daughter. All of the fabrics are cotton, but for some reason, one fabric has frayed so badly that all the blocks using it have frayed right out of the seams. At first I thought some of my seams were too skimpy, but looking closer, it was only this one fabric and some of the fraying was a full 1/3" or more, so beyond the seam line.
To repair it, I am folding the frayed edges under using fraycheck and letting them dry, then I am using a hand stitch to seam the edges back together. If I were at home, I would definitely use iron-on fusible for reinforcement, but have none available here.