Rob, the park is closed from Oct till March so those months are out. There is a local Atlanta club called the Georgia Bounty Runners that support the trail system and might be able to help you out about going out there.
I am also president of another club here in Atlanta, but Beasley is just too hit or miss for us because of the extreme trails conditions.
if you are coming up from FL you should check out Durhamtown - GA - Welcome to Durhamtown
they are South of Atlanta and have camp grounds and such.