Atlantis-I like your nickname....do you sort of feel like the lost city of Atlantis with regards to what you've lost in your marriage? (as an aside)
So, I started reading your OP the other night, but finished it today. Wow. That is a LOT you and your children have gone through.
I'm not here to say if you should stay or go. I cannot possibly advise on this, really, a few thoughts I've had are if he is truly in recovery and the wounds can be healed, I think you've got a chance at a good marriage.
Many married men and women cheat in their minds and in their hearts....they just don't do the deeds in the flesh. This brings up a whole other subject I'm aware....but there is a saying among committed couples: "You can look, but you can't touch." However, some spouses feel they have been cheated on already if their spouse so much as has fantasies about someone else. Then, there are emotional affairs that take place without or without having a physical relationship. If one party feels betrayed....there's a lot of healing that needs to happen if they are to remain together and reach happiness again.
Can you forgive? Because if you are not able to forgive, resentment will rule your heart, mind and even your soul. And resentment can "seep" into all the cracks and crevices of our lives including sometimes our physical health. Resentment destroys intimacy, it really does. And so, if you cannot be intimate with your partner, both you and he will feel a lacking and likely will not have fulfillment.
People can do lots and lots of things to stay together "for the children"....but what about YOU and your needs? I'm afraid you might always be wondering if someone else out there could love you better.....and he will feel your burning resentment no matter how much he tries to make it up to you and repair the damage that's done.
Give all this advice some looking over and remember that no big decisions have to be made right away.