What is a good survival game for console?

I love survival games. I play on Xbox 360, and DON'T have Xbox Live. I'm looking for a game similar to "The Forest" of "Rust," or even "DayZ." I heard about "State of Decay," but I don't have Live, so… I already own Fallout 3 and New Vegas, and Minecraft and Terraria, but those aren't really the kind of games I'm talking about. I mean where you can build and craft things, but the graphics are modern 3D graphics. With combat, and where you have to hunt, build shelter, find resources, etc… Basically "The Forest" only on console and in a retail version. I have searched far and wide, but found nothing. And I don't mean something like "Far Cry 3" or "Left 4 Dead." I mean an open world game where every minute is a fight for survival against the elements, enemies, etc.