I just wanted to say I'm sorry for what you're going through skat. I experienced almost the exact turn of events you have described. Husband distant, moody, out of the blue turns round and says he doesn't know whether he wants to be with me any more. That was the giveaway for me, I know what he's like and there's no way he would ever be able to make a firm decision because he knew he would look like a right bastard if he left me and our small child.
Anyway I too snooped, found emails to suggest an affair with a colleague (and he was only ever at work or with us too, but it's amazing what can be achieved during a lunch hour or "rush hour traffic"!).
I am by no means a strong person, but I confronted him with undeniable evidence and told him to come home and ps k his bags. OW didn't want him, she was younger and not ready to be a stepmum or get involved (oh the irony!) plus they would've both lost their jobs as they work with vulnerable people.
Anyway, as risky a strategy as it is, I do think a short, sharp shock is what is needed here (if indeed he is having an affair). Make him experience how life would be without his family, having 2 kids to stay over in what would likely be a small bedsit or house share. Doing his own washing etc, having no-one to come home to if an evening.
If he is allowed to hang about and casually make a decision it puts him in control of your destiny, plus he gets the best if both worlds. He needs a wake-up call, yes life is shit sonetimes, having small children is hard, but look at the bigger picture. I remember wanted to shake my husband, trying to make him see that this wasn't what he wanted. Who would he spend Christmas with, go on holidays with, what about in 20, 30, 40 years' time?
And if he doesn't want to be with you after all, you have not wasted weeks/months in limbo, walking in eggshells in your own home, worrying that any small thing you do may tip the balance.
Do this and I am certain he will come to his senses, if that's what you want. As it happens, when my husband moved out for a bit I made a real effort to do stuff for me. It actually ended up being rather a nice break! I would have complete peace on a weekend (when he had kids), freedom to choose whatever I wanted for dinner, not having to do as much washing/ironing as before.
If you can get some RL support I would really encourage you. So that you know if he DID leave, yes it would hurt but you would still have people who love you, you ARE loveable, you would still have a life. My family were a godsend, babysitting so I could go to the gym or out for dinner with a friend. I know this is not always possible when you have a small baby, but even if you could get an hour to go out for a walk or a coffee it might clear your head.
Anyway I am getting ahead if myself, this is all ASSUMING he has been unfaithful and then that you want it to work. This happened to me 18 months ago and we had counselling jointly and separately, I can't say we are happier than ever because sadly, in my opinion, he broke what we had and even if it was a cry for help, he should've just cried for help instead! But it will take more time I think, I know I need to be more forgiving (for my own benefit, as much as for his) and we both need to work on some of our behaviours. Although what he did was wrong, I do need to look at how I behaved to make the marriage unhappy too.
Sorry for the long post, I hope it has offered you some comfort though x