Never heard him in person but I think he does naturally have a bit of a cajun accent. But, he probably does turn it up for affect at times. But, it's also possible that the heavy accent is how he grew up and sometimes it just pops back up.
I know that I grew up with a fairly strong accent that I have mostly gotten rid of, but it does pop up from time to time when I'm around others that have it or when under stress.
I don't think he's like Senator Kennedy who went from having no accent to talking like Foghorn Leghorn.