NOLAEMT - "Basically, how do I best get rid of these things?"
About 30 or so years ago, my brother and three of his friends owned a very nice private duck club in Arkansas. Along with it were 160 acres of land. They were having the same problems you have with beavers daming up several creeks running through the property and flooding the fields where rice and beans were planted. Also, they were damaging the levees that protected other fields.
At the time, in Arkansas, beavers were so abundant and doing so much damage to trees and land, etc., that they were declared "vermin," and allowed to be killed 24/7/365.
My brother and his friends bought a bunch of dynamite and began blowing up their houses and dams. Worked for awhile, but the beavers, "busy as a beaver," returned and immediately rebuilt their houses and dams. This went on for some time until the cost for dynamite and effort, became too high... and somewhat dangerous.
So one of the members came up with another plan. He had some relatives in southern Louisiana who lived out in the swamps in prime alligator country. They'd go down there with an enclosed truck and pick up several alligators caught by the "good ol' cajun boys" out there in the dense Louisiana swamps. They'd release them on the duck club property and pretty soon, their beaver problem would be solved.
Sounded like a great idea... until my brother asked him, "So what do the alligators eat after they've cleared out the beavers?? There's not much else around for them to chow down on"
"Uhhhhhh, our dogs, I suppose... and then us, as we wade through the pin oaks shooting ducks."
Fortunately reason reigned and they decided to deep six that idea and just hire a couple of professional trappers to clear out the beavers. This worked, and it wasn't long before everything was back to normal and they had great duck hunting again.
Therefore, I suggest you hire a couple of professional trappers to get rid of your problem, so long as it is allowed by Ga. F&G.
L.W.