I have a couple ideas.
First, dump all of them out, thoroughly wash them, let them dry completely overnight. Then, put a few oyster crackers in the bottom of each one and then refill them with new parm. To dry, I would dry them as much as possibly by towel, then take a pan, put a clean dry towel in the bottom, put a wire cooling rack on top of that, and put each shaker on it upside down and then fill in around the edges of the pan with the lids and let sit overnight. If you set your kitchen heat back at night, you might disable the setback for the night you dry to keep the kitchen warmer overnight.
second idea is to store the shakers in the fridges and get them out as people wanted them.