Do not speak to HR. They are never for you.
I called in sick two months before I started rehab, and it was two months of the heaviest drinking in my life. After rehab I realized that my job was making me sick for so many reasons, and left it. Now on another career entirely after a long time not working at all.
Whatever works for you, sobriety is #1. Possibly better in the long run to get time off from work rather than work through early sobriety. In California (and this may be nationwide) you can contact HR to say that you're taking time away for medical reasons, and forward them a note from your doctor mandating time off. By law they CANNOT ask you for the particulars, nor should you volunteer them. They also can't retaliate or they're setting themselves up for a monster lawsuit due to the Americans with Disabilities Act.
Whatever you think will work better for you and get you sober.