We were just there last week for 3 nights on our way down to Florida. We have stayed at Rivers End Campground on Tybee Island, just outside of Savannah. Its nice but TIGHT spots.
We were trying to stay at Redgate this time around, but thy were booked. We then tried for Skidaway Island, but they were booked. We ended up staying at Savannah Oaks. Let me tell you, the reviews there are pretty bad and I can't understand why...
The bathrooms were clean, showers were nice and hot, park was pretty with lots of Spanish moss. Sites were pretty tight, but the park was only half full so we weren't cramped at all. Nice campstore and even gas pumps right on site. It was about 15 minutes from downtown (1 or 2 exits north on I95). I would stay there again.
Wherever you stay, you need to go to the Breakfast Club on Tybee Island for breakfast one morning. Its worth the drive...