I think that a sway that is genetically part of his build is a bit different from one that develops over time from bad saddle fit or bad riding. But, there are things you can do to help support him, such as being certain that you have a good saddle fit, riding correctly, working up and down hills, working over trot poles or cavaletti.
Things that actually strengthen the abdominal muscles, since they are what supports the back
Enough stuff to keep you occupied for years! not to worry about doing it all at once, though. you are just getting started.
Oh, I forgot an important part: fun. be sure to have fun!