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For my research I dry them under weights. But they get kinda brittle. For our teaching ones we've actually laminated them!
So procedure is: remove from squid, rinse in fresh water, place between folded paper towel, once all gladii are in paper towel packets, I make a pile of paper towels than place the gladii towels on top then another pile of towels, then a board (cutting board or similar) then weights (I used four 5lb dive weights, but heavy books would work too) leave for a week. Mine are stored in plastic sleeves in spaghetti jars, but they can be laminated at this point, they go rather flat but the general shape is preserved.
We also let kids take home the beaks, they dry a treat and we put them in hospital piddle pots so they don't get damaged.