It is illegal to drive in a Texas waterway for sure, and obviously no trespassing. I don't exactly remember how it works, it's been awhile since I looked into this exact same thing. But any and all public is owned by the state, and I am pretty sure no motor vehicles are allowed on unless otherwise stated. not 100% on the last bit, but I think thats how it works with public property
If you ferry over to Crystal Beach, which is pretty neat, you are allowed to drive on the beach down there with a permit.
I don't remember who, but I remember someone talking about there are dunes over near High Island
I never looked into that but you never know.
There is DSO or Down South Offroad near Baytown of somewhere down south
I haven't been to that either but I have heard of it.