New Orleans has well-documented accent variation by neighborhood that is ancestral and not from new arrivals. Also, pretty much any of the culturally Southern metros that have doubled in size since 1965 will have substantial accent variation by default. But if Houston, Dallas, Atlanta, Charlotte, etc. stopped growing rapidly, they would likely converge on a single local accent by the late 21st century, so I'm not sure they really count.
New Orleans is/was a port city with quite a few immigrants coming through, Italians , Irish, Germans, Slavs and even Islenos (Canary Islanders). And, of course, the continental French and Cajuns were in and about the city. The Italians became the dominant backbone of working class whites and their "Yat" accent remains most prominent and basically sounds like the lost tribe of Brooklyn with some locally added vocabulary.
There also remains considerable differences in African-American accents in the city, with lighter skinned (Creole) having distinctive speech patterns compared to country delta African Americans who migrated to the metro as agriculture jobs dried up. Personally, I played basketball in high school and the Creole A-A were easy to understand, one spoke whiter than I did, and one delta A-A teammate I played with for 4 years and still could only understand half of what he said.
