The fact that your cousin's husband is now willing to go to church should encourage her, and I sincerely hope she won't listen to those urging her to give up - both for her sake and the sake of her husband.
After all, her husband (like any other addict) is in the grip of a very strong master - in his case, alcohol. He needs God's help, and he needs the help of other people to be freed from his slavery to alcohol. His willingness to come to church may well be an admission on his part that he needs help - which is the first step in any addict's recovery. He may even find others in that church who've traveled the same road he is on, and can help him gain victory over his problem.
Pray for your cousin and her husband; the Bible says that God "is able to do immeasurably more than all we ask or imagine, according to his power that is at work within us" (Ephesians 3:20). Urge her also to speak with her pastor frankly about this; he will know what resources are available in their community to help her husband overcome this problem.
Most of all, pray that your cousin's husband will yield his life to Christ. Christ will give him a new purpose in living, and also give him hope and strength for the future. It may not be easy - but God loves him, and Christ can change his life.