It is the common snapping turtle ( chelydra serpentina).There is only one other freshwater turtle of that size in Georgia , the spiny soft shell ( apalone spinifera) but the turtles we can see have long necks and are brown,both features of the common snapper (whereas the spiny soft shell is olive green and has no distinctiive neck).
It eats 'anything', according to the University of Georgia's museum of natural history website, which means plants, amphibians, reptiles, mammals and fish.
It's not the friendliest of beasts. It gets its name from its habit of snapping at humans who get too close to it. The ones we see are evidently not close enough to be an annoyance, not that the University indicates that they present any danger!
The American alligator (alligator missisipiensis ) is found "in the sloughs and swamps of Southern Georgia", they say. That description doesn't seem to fit Augusta National, and a Sports Illustrated description of the club and course which includes a list of wildlife there, mentions e.g the red tailed hawk, fish, and snakes but doesn't mention alligators. They'd merit a mention: imagine the litigation if a dangerous animal hurt some visitor or player !(The local University, a touch unnecessarily, cautions against approaching alligators)