Ordinarily, grandparents in GA can't override the natural authority of the parents to obtain custody or visitation rights re: grandchildren unless (1) the g-parents' child is no longer in the picture re: the g-child (e.g., in prison, death, divorce, etc.), and/or (2) the g-parents assert their desire as part of another legal proceeding involving the parents and g-children. Unless one or both of those general categories is applicable to your situation, you may consider a deprivation or termination of parental rights action in juvenile court. However, if you seek legal rights in juvenile court, vs. one of the previous options, just because the mother may not necessarily be a "model" parent is not sufficient to strip her of some or all of her rights. Therefore, regardless of how you approach it, if at all, you will need to present sufficient evidence that allowing you some or all visitation, custodial, parental, etc., rights is in the best interests of the child; how the mother lives her life and/or the health of the child may be factors, but you will need something substantially more than what you have stated to potentially succeed.
Grandparents, even with imperfect parents, rarely win custody. Even visitation is not a sure thing. But both happen in some cases. Far more facts are needed to know what you can or should do, and you should sit down with a lawyer, and share details. Bear in mind that you also have to address the father's rights, which may also be superior to yours.
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Visitation would be much easier. Even if the mother is horrible, you'd have to show the same about the father if you were trying to get custody. See a local domestic attorney.
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