If he wasn't paying the rent, you'd have wound up homeless anyway.
My guess is that he thinks he's "calling your bluff"--that in the end you will beg him to come back, once you've "come to your senses". I say keep rolling with what you started. Getting someone out of the house is often the hurdle that keeps us in bad relationships for too long. You've got that part out of the way.
You will find a place to live that you can afford. It might not be as good as what you had, but no house is a home when it is filled with the chaos caused by drinking. A peaceful house that is tiny and cramped is better than a palace that's in a constant state of uncertainty and unpleasantness.