Low Country weather in September and October generally remains warm with less humidity than during summer months, and is very often still swimming weather. Weather is of course unpredictable - but the weather often comes and goes quickly around Kiawah so rain could just mean a few light showers.
See below for some favorites: Gibbes Museum of Art, Historical Houses, the S.S. Yorktown (fascinating tour as is the submarine adjacent to this air craft carrier), and downtown walking tours.