As someone who has driven several times in the British Isles, I will tell you that it has never been driving on the 'other' side of the road that has bothered me. As Dave and Lifey (two other people with experience on both sides) have pointed out, the fact that you will be on the 'other' side of the car as the driver, and all the other cars will be on the 'other' side of the road as well will make the transition feel natural in very short order. There are also a couple of things working in your favor that haven't been mentioned yet.
First off is the state of the roads here. I spent a week last spring in southern England and was amazed at how narrow the inter-city roads were, especially the B roads. They, and the lorry drivers who insisted on going full speed (and then some) down these roads with no shoulders and no center line were what made driving uncomfortable, not being on the other side of the road. You will find no such roads in America. The vast majority of the roads between your stated stopping points will be motorway quality (Interstates or "I" roads). "US" roads, especially in the south also tend to be four-lane divided highways. State roads will more often than not be only two lanes. But even so those lanes are wide enough for American cars of the late '50s, huge behemoths that make even today's "full-sized" cars look like minis.
The other thing that will be working in your favor is the fact that almost without exception, rental cars in America come with automatic transmissions. So that is one less thing you will have to handle as the driver. Basically there will just be a 'go' button (the accelerator) and a 'stop' button (the brake) and that's it. If you take a few minutes before setting off from the lot with your new car to adjust the seat and mirrors, and familiarize yourself with the other major buttons (lights, wipers, audio, etc.) you should feel very comfortable in short order. By the way, the hire firm will always have somebody available to show you the controls. After all, they want the car (and you) back in one piece too.
Finally, on the one-way drop off fee for picking the car up in Atlanta/Nashville and leaving it in New Orleans, your travel agent probably knows this anyway but often if you make your reservation through a European consolidator such as Europcar or RentalCars.com, that fee is waived.
In short, you are in for a very enjoyable trip. Don't worry. Be happy.
AZBuck