You can't get a bus directly from Destin to New Orleans. Greyhound offers a bus service from nearby Fort Walton Beach to New Orleans. If you want to catch a morning flight then the 9:20 pm bus will get you there by 5:30 a.m. Go to greyhound.com. The lowest fare was $ 36.00 if purchased in advance.. I don't think megabus serves the area but it might be worth trying. Megabus.com
well not use to buses there always a car, but greyhound as above should do it.