Hey Thomas, You are not far from some good pay to play locations. Take I-40 east to us 221 south (marion exit #85) about 7 miles south of 40 on the left are 3 places to go, The Lucky Strike mine, Vein Mtn gold and emerald mine and Thermal City Gold Mine. All these places have camping and are cool places to go.
As far as GPAA claims the closest one to you is the Coker Creek location in SE Tenn, in the Tellico plains National Forest district.
There is a local GPAA chapter at Kannapolis,NC, you might want to check with them and see if they have any places to go. (local chapters sometimes have places to prospect that aren't GPAA claims). It couldn't hurt to join a local chapter, it's lots of fun and you can meet great people.
Other than that you can try the National forests, like Pisgah Nat forest or Cherokee Nat Forest. It is pan and shovel only (no sluices or any motorized equipment). But not National parks, like the Great Smoky Mtns National Park to the north of you. National Parks, Wildlife management areas and most state parks are off limits to prospecting.
I hope you find some places to go, and it's really cool that your wife and kids are wanting to get into it. It is a great way to enjoy the outdoors and meet other folks, especially if you join a local chapter.
Keep in touch, and maybe we can meet-up sometime, as an LDMA member, I go to Vein Mountain Camp occasionally (different place from the Vein Mtn mentioned above) ( I can take guests for 3 day trips), and if you ever want to come to GA ( I live just outside Atlanta) and try your luck, let me know. Pm me if you need more info, I'm always glad to help and share any info I can.