its been more than 10 years but I'm pretty damn sure Pine Lake had a few lean-to's around it. It's a very short drive for you and not a bad hike back in either.. you should be able to make a hike in before you drag your family and gear in, I think you'll like.
another great spot we used to sneak the atv's into off the same seasonal road as above. A much larger lake, but you may have more foot traffic, no idea if atv's are still sneaking into it though. Its a gem on the west side.
If you're willing to hike in, there are endless options for both campsites and Lean tos. DEC also has a ton of what I call "car camping" spots, most are not published anywhere (a good resource for these is andyarthur.org, just search the area you are looking and he has a lot of maps). If you are willing to hike a little, Bear Lake Lean to near McKeever is in my opnion, one of the best lean tos in all the ADK. Amazing spot and easy hike. If you want vehicle access, Woodhull Lean to (same area) is one you can drive right too, its on a seasonal access road, however, which I don't believe opens back up until May 15. The road to Woodhull lean to also has probably 5-10 "car camping" spots along the road, some nicer than others.
I've been part of the ADK Mtn Club Lean to adopter program for a few years, the one I have is in Lake George Wild Forest, I'd highly recommend this area but its on the other side of the park from you.