If we could see a complete photo of the quilt top, it would be easier to suggest. You may be able to reattach the pieces. Looks like your 1/4 inch seam may have been closer to 1/8. Full shot of the quilt would help. How old is DGS? That makes difference too.

I've only made one quilt and didn't have that problem but I did have some strips that didn't line up properly so I appliqued a patch over that area. Since it was a baby quilt and the original panel had some butterflies on it, I decided that the extra butterflies needed to fly away into the border area. Just covered those misaligned seams right up. This would surely cover up your frays but not sure if it would prevent your seams from further fraying during future washes.

What about ric-rack over the frayed places? Calla

This is one of the reasons that fabrics should be washed before you start. Also takes care of shrinkage and colors bleeding. I can assure you, you won't make this mistake again.
Sorry this happened to you but don't give in to it. I wish you well.