I agree with Kirk Rush's suggestion for the Charleston Museum. If I could only visit one house museum in Charleston it'd be the Heywood Washington house, no hesitation. You'll see several Thomas Elfe pieces, as well as the spectacular Holmes-Edwards bookcase. The Nathanial Russel house is probably next on my list, but more for the architecture than the furniture. The houses on the battery are newer and tend to be filled with Regency furniture, if that's your thing (it's not mine).
Be sure to explore the antique shops on King Street.