Hi, the known donor registry has a listing of the applicable laws for each state. here's Georgia's:
It looks like technically it would be a felony for you to do 'artificial insemination' if you're not a doctor, but I'd be surprised if anyone ever called you on it. This looks like it's trying to keep people from setting up illegal clinics, not stopping individuals from doing whatever they are going to do in the comfort of their own homes. Might be worth asking a lawyer about--this is a question that they could probably field for free, or that legal help line. I think your issues are going to increase if simple at-home attempts don't work and you need to find doctors to help. This clinic: http://ivfga.com/services/lgbt
advertises that it helps lesbian couples, so that might be a start--if you're not near them, they might be able to recommend someone closer.
Unfortunately as Pokey says, there's no way you'll both be listed as parents on the birth certificate if the baby is born in Georgia--you'd need to have them in a state that has gay rights protections (either sees you as married or allows same-sex parents to be on the birth cert) for that to happen. If that's really important to you, and it's financially feasible, you could consider temporarily relocating to an equality state for a month or so around the birth, but that's pretty extreme. In the same vein, there's no way that a known donor father won't be considered the legal father in Georgia, if it ever comes to that. The times when that seems to become an issue for folks is when either the known donor decides that he wants parental rights and sues for them, or if the moms need to apply for assistance--there was just a case in Kansas where they determined the donor to be the father against everyone's wishes for the purposes of garnishing child support payments. The only way to avoid that is to use an anonymous donor, which isn't an option for everyone.
I think that most people in states that don't recognize their relationships do a series of legal documents (powers of attorney for property and healthcare for each other and the baby, papers indicating guardianship in the case of incapacitation of the bio-mom, wills, etc), talk things through very thoroughly with their known donors to bring up any red flags before they get pregnant, and hope for the best. There are lots and lots of queer couples raising kids in the south (and other less-than-gay-friendly parts of the country), it's just a little more legally precarious than if you were in an equality state.
Good luck sorting all of this out! Hopefully soon all of this will be cleared up at a national level and none of us will have to jump through these stupid hoops just to make a family.