Hi Nursey, this reply is probably a bit late as I've only just seen your post, but when I read it I found myself thinking a lot about your predicament. I'm certainly no relationship expert, but I can't help thinking that neither of them is ideal for you. I know that's probably not what you wanted to hear, but:
Regarding the children's father, I can fully understand where you're coming from when you say you feel you should let your children grow up with their two parents. In fact, initially I was going to say that this is the option you should take. However, the fact that he cheated on you when you were pregnant, not just once but both times, speaks volumes. If you still feel bad about how unhappy he is now, do bear in mind his behaviour. Can you be absolutely sure he wouldn't do it again, a few years down the line? If he did, the effect on your children could be devastating.
Regarding the girlfriend, hmmm... Well, she hasn't exactly been faithful either, has she? She got into a relationship while she was already seeing you - I know you described her as a friend at this point, but she was still sleeping with you and so must have known that her getting involved with someone else would be hurtful for you. And the woman she started seeing - did she know that her girlfriend was seeing you as well, or was it all behind her back? Your friend wasn't being fair on either of you. You've also mentioned that she has issues, and that she blows hot and cold. So it's not always great being with her, but you're perhaps looking back on the whole thing with rose-tinted spectacles a little bit. I can't help wondering if, when the passion dies down, the relationship will remain happy. You say you have cut all ties with her - there must have been a reason for that?
Don't hate me for suggesting this, but could you perhaps be going through a period of loneliness? Being a single parent can be pretty solitary, you have only two young children for company at home and perhaps don't have much of a social life as you have to look after them and so it's difficult to get away. You don't say exactly how long you've been single, but it seems a bit odd to be longing for someone you consciously cut ties with. Is it even still an option to get back with her, is she still single, would she want to see you? I guess another possibility is simply that you are gay, which is why you think of her and feel nothing for the man who wants you.
Ultimately, I can't tell you what to do, nobody can, but I think if you are at all unsure, then remain single for as long as it takes, don't jump into anything. If you do decide to get back with the children's father, the two of you would need to have a serious discussion about his past behaviour. He's only human, we all make mistakes, but to have done that when you were pregnant - twice - is really low. I hope you make the right decisions and have a fabulously happy life, I wish you the best of luck. X