Fayetteville NC to Gulfport MS

If you take I-95 to I-20 and then I-20 to Atlanta, the route will take you for just a few miles on I-285 and then I-85 to Montgomery and I-65 to Mobile.
Then to I-10 west.
You really won't be in any of the 'Atlanta Traffic' that everyone is so frightened of. All of that is downtown and on 75 and 85 coming in from the north. I travel the I-20 route to I-285 very often and although it can get heavy in the early mornings, after about 10 AM it is just another drive. Same with the sort section of I-285. Just watch the signs and follow them to 85 SOUTH. Take the wrong exit and you will be 85 NORTH and headed to downtown.
It is sort of interesting to drive under an active runway and taxiway as you pass the airport. Something like 10 lanes as you do this, so no worry.

Atlanta traffic can be bad almost any time. If you can't stand out I would go the other route.
Another alternate I is to go I20 toward Atlanta get off in Greensboro, GA head to Macon then take US80 to columbus then to I85. That way will take longer then going through Atlanta but less time than going down I95.

Here is a route you may want to consider:
95 from Bragg to Florence
20 from Florence to Meridian MS (Go north 285 at Atlanta then through Birmingham)
55 from Meridian MS to 10
10 to Keesler, if that is where you are going.
I live in Bham and this is the way we go when going down to Biloxi and stay at Keesler. I am also retired SF. 5th, 1st and SANV.