Oh no! I'm so sorry to hear what has happened. I've never heard of them going through plastic containters.
I know it wouldn't be the same as from the family pieces but there are places to get old fashioned hankies. Maybe you could get some newer ones. I seem to remember seeing some in a catalog somewhere. Also what about sending out requests to any of your other quilting friends and guilds. Check with some local knick knack, antique and thrift shops. Put a request out on craigslist asking for old hankies. You never know what response you might end up with. You could then add the few pieces from your family that you have left into a special part of the quilt, the center, part of the label etc. Just a few thoughts I had that I wanted to share. Good luck with your search of some replacements and your quilting project.
Pam M