Keeping the need for my sabbatical secret is not realistic. Finding a replacement instructor and delays in my research will cause inconvenience to multiple parties.
What steps can I take to reduce embarrassment and protect my reputation as much as possible? This request will surprise most people in the department.
This is a tough question, and you have my sympathies.
A sabbatical is time off from on-campus responsibilities given to enhance your long term scholarly life. It varies in its implementation from place to place (and in fact they do not exist at my institution), but most commonly you propose travel and research plans.
Alcoholism and mental health issues are two kinds of recognized medical problems. It is my understanding that you do not take a sabbatical because of medical problems; rather you go on medical leave of one kind or another. Trying to hide medical problems under cover of a sabbatical sounds, at the least, to be very risky, and you seem to recognize that this is not practical in your case. So I would recommend that you look into going on medical leave instead of sabbatical.
Treatment of either of your medical issues will probably involve a psychiatric / counseling component. I am not a psychiatrist or a counselor (rather, I am a mathematician: really not close!), but I would humbly and tentatively suggest that even before you go on medical leave you seek some counseling. In particular you mention the embarrassment of being sick in the way that you are. I think it is completely normal and understandable for you to feel embarrassed / ashamed about these problems, but those kinds of feelings could lead you down very unproductive paths, so I suggest that you concentrate some of your earliest counseling sessions on dealing with that. In particular, a counselor probably has very helpful things to tell you about disclosing your problems to others: whether, when, how much, and to whom. As far as I know, it should be possible for you to get the same medical leave and considerations as someone with less socially stigmatized medical problems without having to disclose the details of your medical issues to anyone in your department, unless you want to.
I wish you all the best. I hope you do take medical leave and get knowledgeable, helpful care.