My kids are older than yours-16 and 18.
I didn't tell them immediately but have been honest with them since I made a real commitment to stay sober.
They have seen the ugly side of alcoholism from my brother and understood a bit anyway.
I told them that drinking was making me ill and although it didn't affect me in the same way as their uncle, that didn't mean it was any less of a problem.
I told them that I go to meetings to learn about alcoholism and to support me to stay well. They accepted it without question.
Think kids need educating about alcohol anyway and you're in the best place to do it.
Good luck x