I don't know how things "work" in your family, but I don't see that everyone in the world needs to know every issue you have. I DO think it's unfair and unreasonable not to discuss it with your husband. He is the person you should be closest to, and sooner or later it's likely to come out, and he might feel very hurt and betrayed that you didn't confide in him. I think it's totally reasonable to ask him to keep the information private, at least for now, and until you feel comfortable disclosing it to someone else.
It's usually very helpful in early recovery NOT to keep alcohol in the house. Without discussing it with him, he might not realize he's creating problems for you by drinking around you.
If your drinking is not YET at a point where you are endangering your child, by addressing your problem now, you are being the ultimate in responsible parenting. I work in the legal field, and it's next to impossible to prove someone is an unfit parent, even when they have untreated addictions. Unless you have abused or neglected your child, there is no basis for anyone to say you are "unfit."
Please consider having a heart-to-heart with your husband and ask him to support you on this. At some point, other family members might notice you aren't drinking, and ask why. All you need to say is that you've decided you feel better when you don't drink. Period.
Please don't try to do this in complete secrecy. Having a drinking problem isn't something any of us chooses. But the sooner you address it, the happier and healthier you and your family will be.