Why not look in Warrensburg, which is about 5 miles north of LG on Rt 9. There are also several CGs near Lake Luzerne which is 7-8 miles SW of LG on Rt 9N.
Both of these areas are just far enough away from the usual 'zoo' of Rt 9 and the 6 Flags park (Great Escape) or Rt 149. It is the summer flat track horse racing season at Saratoga, 25 miles south of LG in August so any campgrounds will be extracting as many dollars from the unwary as they can.
We pass through the LG area at least a couple times a week between May and October on our way further north but try to stay on I-87 unless the DW feels the need to shop at the outlet mall off I-87 Exit 20