Sean--My knowledge of the area stems from the fact I worte a book in 2005 called "Golf Charms of Charleston."
If access is no problem, the privates are far better than the public/resort courses, in most cases. Yeamans and CC of Charleston are the two "old guard" Raynor courses. The nouveau courses are (in order of preference) Cassique, Bulls Bay, The River Course and Briar's Creek tied for 3rd.
As for Kiawah, the Ocean Course is a must-play, followed by (again, in order of preference) Cougar Pt., Turtle Pt, Osprey Pt and Oak Pt. Lesser lights in the public sector include Stono Ferry, Snee Farm (rotten conditions, unfortunately) Dunes West and RiverTowne. As for Wild Dunes, the Links Course is a shadow of what it once was, and the Harbor Course is a joke. As JNC pointed out, two decent underrated courses are just adjacent to Kiawah on Seabrook Island.
Have a good trip!