After installing three new 537 servos in the servo tray there is an obvious difference in the amount of noise each one makes as it turns back and forth. One is way noisier than the others. Could it be the screws holding them to plywood is causing a distortion in the case and therefore to the gears?

You may have one servo touching the tray or the screws my be a little too tight. If you're worried about it, pull them out and plug one in at a time while holding it in your hand. See if there is a difference in the way each one sounds. Do the same test while holding the servo against the table top and it will sound much louder.