Ohio is definitely the home state and has jurisdiction over all matters of custody, visitation, and in recent years child suport as well but that may have changed. You should be awarded temporary custodial parent along with orders to the non-custodial parents to turn over the kids to you at a date and time you decide. By
court order. In all ways, legally, ethically, morally, technically the mother has kidnapped the children (especially since she sought no legal remedy and her hands are dirty). Georgia has no jurisdiction but could gain jurisdiction if you don't file in Ohio.
Normally fathers are treated much worst in out-of-state courts and if you are served with any kind of papers in Georgia without filing in Ohio might be enough for substantiate Georgia jurisdiction. You and the kids are very much the injured parties since the children have been denied access to their father. The mother needs to know the
child support/visitation systems were not intended as proceedures to sell children.
Needless to say, the reason you have not have access to the children is understandable and common sense, so don't let parties from either state play that tired old tune.
You very much have the upper hand so keep it and play it for all it's worth. Your children will benefit in the present and future.