When I kicked out the last guy I lived with (not an alcoholic, and we had already broken up but he was dragging his feet about moving), I said, "That's it, you're out of here today and I'm staying home to make sure you do it." When he said, "But I've got no place to GO," my response was, "Not my problem."
And guess what, he FOUND someplace to go. He came back with his son a few weeks later to move a few items he couldn't take with him that day, but other than that we were done.
You've got more power than you think.