I've always paid mine on or near the due date. If I have the cash I leave it in my account earning a little interest (key word little). If using a charge card for reward miles etc. I wait to the last minute to avoid interest charges. I've heard paying it in full puts you in a less advantageous position for further price reductions on some cruise lines. I don't specifically recall which ones, or even know if that's true.
As the others have noted, the penalty phase starts with the final due date. No advantage whatsoever to paying it in full in my opinion until you have to.