First of all, minimal handling. Secondly, starch the fabric before cutting (but then, you said you already do that). This is what I do to avoid handling the cut pieces too much: As I cut the pieces, I stack on something that can be used as a tray to transport the pieces from place to place. (I cut on my dining table and carry to the sewing room where I lay them on a table.) I keep the heavy cardboard squares that come in packages of scrapbooking supplies. These make wonderful trays. I also use them for laying out my block before I begin to assemble them.
Oopos! Seems like we had the same ideas.