Black mold can cause a host of health problems. Fortunately, if you find black mold in your apartment in Georgia, you have some legal options.
www.enjuris.com
Georgia law is vague, but case law isn't. They don't wanna **** with legalities involving that if they can help it, but it depends on how far they want you to go legally to prove that it's now their problem. Often times it will require a legal expenditure on your part before they decide you have the means and the will to hold them accountable. Until then they'll rely on the vagueness of the legality and the likelihood that you have nowhere else to go and/or don't want to go through a lengthy moving process over it.
There are situations where you would be partially or wholly at fault, but from what you describe your case isn't one of them. You told them before it became a problem and now it is. It may require you seeking out what type of mold it is, I'm not sure. That would certainly help you legally, but they should be asked to run that test before you.