Dunster, how lucky to receive such nice tops. They probably were washed at some point and that has resulted in the fraying since there was not the quilting to support those narrow seams.
All the suggestions so far are good. I would do the applique a similar piece over the frayed or damaged pieces. You didn't mention your intentions for the quilts after they were finished, I think provides the most stable pieces once finished.
Not sure what method you planned for quilting but you will want to be careful when you tension the quilt for quilting that you do not pull any more seams apart.