I think you've been given incorrect information. Did you get this from your local public high school? If so, that would be why it's incorrect.
Your mother was well within her rights to quit submitting attendance records when you turned 16, as the law does not require her to do so. And it does not make you a high school drop out.
Your graduation has nothing to do with the school board. The state of Georgia does not issue high diplomas to homeschooled students; the parents do. That means you don't have to take an SAT or an equivalence test to graduate. If you and your parents are satisfied that you have completed your education at home, they may graduate you.
Have you taken the college-entrance SAT (Scholastic Aptitude Test)? If not, you should check into doing that (it's different from the Stanford Achievement Test usually given at the end of each school year through 12th grade).
No, you don't have to take the GED, either.
You should check with a local homeschooler. Here is the link to GHEA
. You should be able to find someone who can answer your questions.