I've never lived anywhere with Armadillos though I sure saw a lot driving from TX into Memphis. Do they have any natural predators or are they afraid of humans or domestic like dogs or cats? Reason I ask is we got rid of a burrowing rodent problem years ago by sticking balls of cat fur down in their dens.
A fence is best. I like them and think they are cool, but they are very destructive. Here are some things you can try.
You can add a border using logs or bricks to stop them getting in.