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While pre-cruise touring is not generally recommended, luckily, this is quite feasible in New Orleans as the Port of New Orleans is just a couple of blocks from the French Quarter.
Barring unusual circumstances, you should be able to go directly to the Port from the airport and check your bags with the Porter at the pier. Bag check begins well before check-in opens and embarkation begins and you likely won't arrive at the pier until 11am anyway.
For clarity, you check bags with the Porter before you check-in for the cruise.
You can pre-print the bag tags before you leave home.
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This service looks like exactly what you are looking for: they pick up the bags from the belt at the airport for you, and deliver them to the ship.