I live next to a State Park. It sounded really cool when I bought the house. No neighbors! Privacy! There is a trail accessible from my properly that goes into some woods, which you only want to be in during the Fall / Spring because the ticks and chiggers are intense.
The biggest issue with sharing a border with the state is the fence line. Technically it's on them to maintain. The woods are filled with mostly 200 foot trees. They exist right up to my property and fall every year during big storms. There goes the fence line. I have dogs so the fence is important. Otherwise they run out into the woods that have licensed bow-hunting during the Winter.
In practice its impossible to get the fence fixed. You are working through layers of bureaucracy and mismanagement. There is no vested interest from a parks service to work well with hundreds of neighbors individually. I'd build fence lines myself, and could easily pay for it, but it's the upkeep of those fences that would be difficult because so many trees (really, it's every year) fall down from their side.
My main point is that sharing a properly line with the government is more like sharing a properly line with nobody. You may as well own that land. Don't expect any upkeep and go in with eyes open.