My family was just in New Orleans for Christmas. I researched town cars, airport shuttles, Ubers and cabs. For the 4 of us, it was cheapest to take a cab to and from the airport. The cab fee is standardized, so you know what it is before you get in the cab. There were several cabs lined up at the cab stand at midnight on Christmas Eve, so I don’t think availability will be an issue.
We stayed on the edge of the Quarter near Frenchmen’s Street. Our plan was to use the riverfront trolly line (the French Market stop was across the street) and walk. But the river front trolly line was damaged during the storms a few months before and had not yet been repaired. I don’t know if it’s running yet or not. So we used Uber to get around when we couldn’t walk.