I think it is supposed to fray - makes it soft. I made two doll quilts for my DGDs and FMQ everything together. Kids are happy.

I will agree with Tartan, but then I find that I agree with her much of the time. The sheer fabric should solve the problem, a whale of task you have; but other than sewing each section down again it is the best solution.

I looked at the quilt, and the easiest thing I could think of is something I saw once on Nancy Zeiman. It was here appliqué demonstration. She set up for free motion quilting, then went around the appliqué with kind of a loose free hand zig zag looking stitch. It looked to me to be quick and easy since you are not holding them down this time you are just stitching enough to hold the threads to the edge of the pieces. You could aim for sort of a loose lightening look. Just a thought. I wish you success whatever you decide.

Oh brother what a drag!! I often wondered how these rawbedge appliqué quilts would hold up...I guess now I know. Thanks for sharing.