May, go easy on yourself. You are allowed to be as angry as you like for as long as you like, but blame never ever helps. For a marriage to work you need to take 100% responsibility for your part in things and no more, and so does your H. He isn't able to do this. You are doing this as best as you can and you did it as soon as you knew how to. Your husband is failing as a husband and as an adult. You can be furious about that, but it isn't your fault and there isn't a great deal you can do about it right now.
Your core values are keeping you stuck in a situation that is no good for you. I think your core value is to give your girls the very best upbringing that is within your capacity to provide. A happy home in a stable marriage is not on the menu of things you can give them right now and that 100% STINKS. It really does. I can't offer it to my children either, and it does, in my quiet moments, break my heart. I hear where you are coming from. None of that changes the fact that the family that your children are existing in now is fake. I don't doubt you love them and I believe your H does too, but this illusion of a happy family isn't real.
I bet you know children who are growing up between two houses with blended families who are just fine. Yes, it is painful, and no, it isn't what you wanted for them, but I sense a bit of judgement here that kids being brought up by separated or divorced parents are getting second best, and I don't think that is true. I am sure if I end up bringing my kids up alone, with H visiting them in the week or taking them at weekends and holidays, you would not judge me, would not think I was a bad mother, and would not assume I was not offering anything other than a great upbringing to my kids. If there was ever a kind step father in their lives, I am sure you would not judge me as giving them second best. There are infinite ways to make a family. And infinite ways for kids to navigate the normal bumps and lumps and transitions of life.
You have the opportunity now to show them what self respect, honesty, boldness and boundaries look like. If they are girls, this is especially important. You can show them that within a good marriage, or you can show them that in a collaborative divorce. I am not sure you can show them that in your current situation. They might not be aware of what is happening now, but I can see unless you act to change it decisively, this limbo situation will spinning on for years, your anger corroding into bitterness, and your children, as they mature into young women, learning that Mom needs to go along with what Dad wants to keep him happy, no matter what it costs her. They'll see Mom's light slowly but surely going out, and think marriage is about Mom being angry and resentful and unhappy and Dad being an entitled teenager. The more angry and resentful you are, the more distance there will be between you and your H and he will end up in another affair, or just continuing with his current mistress for as long as it is convenient to them. I don't know of that many more damaging lessons to teach young women about heterosexual marriage: I am sure many of us women are here on these boards because, in some way, we learned that lesson from our own mothers.
And when your H says 'we can make this marriage whatever we want' what he means is 'what I want'. He really doesn't care at all about what you want and he hasn't done in years - if he was so interested in having an evolved marriage, he would have involved you in his decision to have another sexual and intimate partner. He kept that secret from you and took away your consent to the relationship you were in. He's still doing it now, only he's doing it to your face.
