This is literally the only decent road I've found near Charleston, SC (with one exception, which I'll note later).
Francis Marion Park is on Hwy 17, maybe 10 minutes north of Mount Pleasant. It's a bunch of dirt roads winding through the forest. The whole park is larger than the entire Raleigh and Durham (NC) area combined, just to give you an idea of how large the road-network is, here. It's what our cars were made for.
I've heard there's good driving by Hilton Head, but I haven't made it down there yet. The only other good place I know of is off Hwy 61. Coming north on 61 from West Ashley, before you hit Summerville, you'll pass under a set of large high voltage power lines. Right under the lines is a dirt road. I have no idea what it is, but I think it's a gun club. It's a series of connected dirt roads, and it's a lot of fun. It's also possibly illegal, so you might want to stick to the FM park.