How about always pulling the sheet of tissue out with the print, thus always leaving image surfaces protecting by a stationary sheet? Maybe put a piece of cardboard in the bottom of each drawer, remove all of the prints at the same time on top of the cardboard, and carefully turn them over on a table until you get to the one you want. Or maybe if you find the one you want, lift the prints above it with a rigid piece of cardboard?
Good luck ...