I really don't want to give you advice about telling your husband about the infidelity, but here goes.
I think that is a separate issue from your alcoholism.
You were unfaithful because, as you state, you were drunk.
I would give it some time. I would put all my energy, if I were you, into getting sober.
I would wait until I had some real sober time under my belt before discussing this issue. Because it could end your marriage.
Do you have a program?