I don't want to give false hope, but I do believe some people get close enough to death that they can stop. There is nothing you can do to get trust back except for him to follow through and time, good clean time. You don't have to walk on eggshells all the time. Have a plan and let that be enough. Keep a wide support system.
Denial is very strong. He has opened up to others about his disease, that is a super good thing. Support and encourage him. Even if he does not want to go to AA, if he can have a friend who is close by who could as as a surrogate sponsor so to speak would be a good thing.
God bless to you and your family! Keep coming to SR, it is a great place of healing and support.