Mathilde, glad it worked out! Just as an aside, my boyfriend didn't fully understand how much of a problem I had, even after I told him. I was a secret drinker, too, and apparently ok at it.
What's been interesting is that after a while I didn't have to try to explain, because the difference between me now and a few months ago is so noticeable. He's always been supportive but now I think he fully gets on an emotional level how much I need this. I would predict your partner will probably grow in his understanding, too.