In NY, the state parks will be heavily regulated having designated pay for/reserved camp spots along with amenities. However, the forest preserves (Adirondacks/Catskills) and state forests do allow backcountry/primitive camping.
Just look for a nearby state forest and look up the dec regulations for that site. More than likely you can can primitive camp there.
I've done it, it's just a huge PIA compared to out west where you just get on forest roads where there are clearings you pull into an camp all over the place.
The east coast just has this strong hostility towards camping for some reason.
There are a lot more people looking to camp in generally smaller areas especially on weekends. And the east doesn’t have the large networks of forest roads and almost roads and not really roads that you have in many western national forests and BLM land.
The West obviously has plenty of camping restrictions in popular areas like national parks and wilderness areas where you may have year in advance lotteries.
I’m not sure why the east would be uniquely hostile given the same agencies administer federal land across the whole country.
But, in general, yes if you want to get away from people and have more flexibility in where you can camp, you’re probably best off traveling to the western states.
There is comparatively very little federal land on the east coast. And even where there is land, it's not as friendly as the west.
Using white mountain NF as an example, their camping policy is much more restrictive than western national forests (i.e camp away from roads and trails).
There is virtually no BLM land and national parks are kinda strict all over. So you are left with state land and they tend to like structured campgrounds that you pay buy permits for.
Just look for a nearby state forest and look up the dec regulations for that site. More than likely you can can primitive camp there.