We like Green Lakes when we want to be close to Syracuse. Some of the sites are pretty close, and the electric sites go quick. The reservation window is nine months - so May is open now. Overall, its a pretty good campground, and it is really close to everything along routes 5 and 92.
Other than that their really isn't is much in that area for private campgrounds. The old Plainville KOA closed several years ago, and we haven't found anything other than Green Lakes in the Syr area. There are a couple public parks (one state, one county) on Oneida Lake. Never stayed at either one. Tried a couple times in the summer, but couldn't get a spot at the last minute.
I keep telling DW our "semi-retirement" job should be opening a private CG near I-81 around Syracuse. There really aren't any private CG's in the area.
2005 F-150 FX4
2013 Rockwood Roo 23 IKSS