Pets are prohibited on state park beaches. They are permitted in the campground. However, unlike the state park, St George Island is very pet friendly and you can bring your pup into restaurants and onto the public beaches.
Not in the state park, but all the other beach areas on SGI are dog friendly. So you'd need to leave the SP and head to one of the public beach access points with the pooch.
They're allowed at camp sites but not on the beach, trails, boardwalks or pavilions, with the exception of service animals. When outside, they must be kept on a leash no longer than six feet.