Yes, it's possible. My wife and I have a large family and are coming up on nineteen years of marriage. I drank, progressively worse, for about 22 years. I'm now over five months sober. Yeah, it's been rough at times, and she, rightfully, didn't trust me for a while, but that trust is being rebuilt with sobriety. In many ways, there is more honesty and open communication in our marriage than there ever was before.
Does he desire sobriety, or is he acting like this is something being forced on him? I sincerely hope that he desires it. Even if he doesn't fully at the moment, perhaps he'll learn to embrace it. Some of us are obstinate like that. I sure was.
I hope the best for your marriage and your young family. I pray for your husband's sobriety and peace in your family.