He does have a key, but he is not on the lease...does he pay rent? Own the place? Or do you? If it's your place, and you are uncomfortable with him being there (which, given his past actions, would be understandable), you have every right to ask him to go to the shelter. It sounds like in your heart and gut you know this is the best thing for both of you.
What has helped me....my therapist suggested I think of my sister as "the alcoholic" rather than "_______, my sister". It helps me to see that she and her disease are one in the same while she is actively drinking and not willing to seek help. She is defined by her disease, and her disease defines her and her actions. It helps me to see that while she is my sister and I love her, I do not have to put up with the crazy madness that is the reality that an alcoholic creates.
He is your son...and you love him....but he is also an addict, and he needs to do this on his own. And you deserve sanity and peace in your life.
Sending you hugs, chicory...stay strong.