Ty Cobb didn't have much of a Georgia accent either in the recordings I've heard, all from after his playing days, though you can sometimes hear a bit of a twang.
From 3/19/30:
https://www.youtube.com/watch?v=rqnH5BrRWnk&t=332s
From 12/1/55:
https://www.youtube.com/watch?v=xYDkBuyveiA
Some people retain strong accents even after living for many years in an area with a different accent, but most people adapt their speech to their surroundings (mostly unconsciously).