I believe the James Island County Park only allows stays up to 28 days which is longer than most State Parks which only allow 14 days at a time. To stay somewhere long term, you need to check out private RV Parks rather than public parks. Private RV parks usually provide smaller, less rustic sites and less privacy than the public city, county, state, and national parks but allow you stay pretty well as long as you wish and will often have weekly and monthly rates at a substantial discount from the daily rates. You normally pay for your electricity separate when renting by the month so keep that in mind when comparing rates.
In Charleston, we have found
Oak Plantation to be a very nice park but I have no idea what their monthly rates are - give them a call. If you haven't tried it yet, you need to get acquainted with the
RV Park Reviews site where you can find public and private campgrounds by location and each listing will also have a link to the campgrounds own web site.