That's the "Cajun" accent. You'll hear that more in West Louisiana, home to the Acadians, or "Cajuns."
Mary Landrieux has a light cajun accent; you can hear it in words like "water" ("waduh"). My wife is from Lake Charles, La., and worked to overcome it. But, when she talks to her family, she slips into her cajun, and she's not even aware she's doing it until I start laughing!
Aww.