Welcome Mom - I see a pattern of behavior in him that is disturbing. HE is drinking too much, he has been unfaithful before, and is engaging in behavior that is questionable at best. (Personally I would call cheating, since he is hiding it from you, and is not a way he should be behaving) He has managed to minimize things, and blow smoke to the point where you question your own personal judgement.
Take a step back and try to look at this as an outsider. Now is his behavior acceptable? Whe you R'd from his first A, did he fundamentally change, or just tolerate the new rules that were put in place to keep it from happening again?
Does he own what he has been up to this time? If not then I say you are doomed for repeat behavior again and again. He has to get what he is doing is wrong, was wrong, and is horribly hurtful to you, your marriage, and is creating an unhealthy marriage for your kids to emulate from.
So great he gets that he needs to do something or he is going to out on his ear. But will it be enough? Only he can decide that.
My recommendation to you is to maintain in house seperation until you really see him doing the hard work. See an attorney and find out what your rights are. Listen if this is a deal breaker for you, then it's a deal breaker, and there is no shame in that. If my H EVER cheated again I would be done, not even question it. He'd find his shit in trashbags, the locks changed, and his toys for sale in the front yard.
Sit down and figure out what you really want, and what you need. Get yourself to an IC, to work toward getting yourself stronger, and finding you again. When you become a mom, and have back to back to back kiddos, you tend to loose who you are, and that's how motherhood is supposed to work, but now it's time to focus on you. You have to put yourself and your kids first.
In addition to the above things, please go get yourself STD tested, just to be safe, and talk with your Dr. about the stress in your life at present time, and get a little pharmaceutical support if needed in these early stages. Sleep, eat, stay hydrated, these are essential to clear thinking and making good choices.
Lastly since his behavior is a bit from one extreme to the other, I would strongly recommend stashing some cash, just in case. You never know what is going to happen, and if you don't have go money as I liked to call it, you should. Enough to take care of yourself and your kids for a few weeks. You never know what he is going to do. If you start playing hard ball, you needed to protect yourself.
((((and strength))))