Cloudland Canyon State Park is my favorite state park in GA that I’ve visited so far. It is in the Northwest corner of the state near Lookout Mountain. We were lucky enough to reserve a dog friendly cabin here for Valentine’s Day weekend 10 years ago and it snowed (obviously a rare occurrence in GA!). Most of the state parks have little cottages to rent and some are dog friendly. This park has many trails to choose from. The Waterfalls Trail is a must! Park day pass 5$, Annual passes available for 50$.
https://gastateparks.org/CloudlandCanyon
Trail Map link: https://gastateparks.org/sites/default/files/parks/pdf/trailmaps/CloudlandCanyon_TrailMap.pdf