The kids did. Archie and Olivia were originally from Mississippi, but made NO their home after Archie was drafted here. Even when Archie moved around from team-to-team in his last 4 years or so, Olivia and the kids continued to stay in NO. Peyton, Eli and Cooper only left when it was time to go off to college. Their accents were well established by that time. All 3 boys were born in NO.
But children tend to pick up the accent of their parents to some extent and I expect that is why they sound kinda like their Dad.
BTW, I grew up in St. Charles Parish and didn't leave S. La. until 6 years ago. But my parents were not originally from there and my accent is like my parents, not like the area. FWIW.