For a non-Western state, missouri has an amazing amount of public ground, especially in the southwestern part of the state. Much of it is in the Mark Twain National Forest and is open to hunting. I said earlier that my first turkey was from there years ago, and it was on public land. I haven't hunted there much recently because I moved back to Kansas and I have access to private land closer to home.
You can certainly get away from the pressure. I never saw any significant pressure once you got off the roads. I've heard good things about hunting around the various corp of engineer lakes too, although I've only tried it once (Truman). I've seen turkeys around some of the others while duck hunting too.