Whether you should just pay the fine or depends on what the rest of your driving record looks like. If this is your first ticket then your situation is vastly different from someone with multiple moving violations in their history. I have included a link to the GA DDS points system webpage to help you see where you are.
Is your case in Elbert County State Court or in Elberton Municipal Court or somwhere else? You need to speak with a lawyer that is familiar with court sytstem in Elberton and Elbert County. I pratcie in Elbert County of a regular basis and assure you that Leaving the Scene charges are not to be taken lightly you can lose your license, get hit with jail time and a big fine. Call a lawyer wih experinece in the area ASAP.
If the children are not his and were not born to his wife while he was married to her they are not his legal responsibility. Based on the information conveyed in your questions, he is not legally obligated to pay child support.
Georgia Department of Driver Services gets the money. Get yourself an attorney that know what they are doing or you will find you have a long row to hoe. After you are in over your head it may be too late.