Hi LittlestHobo, I'm going to take another approach to most of the replies.
Here's what I'm reading: your husband is far from your ideal partner. It is dragging you down. He doesn't show affection, he doesn't give you any sexual attention and uses excuses to avoid it. It's depressing, it's lonely, it's not what you dreamed of. Surviving 12 years of this seems .... impossible but necessary. As the alternatives look worse:you know there are some advantages to staying. Financially it's better. You don't want to lose your home. You are accustomed to dissembling, so the kids havent really noticed you arent really happy and they seem happy enough. You can shield them from his temper better if you stay. You want to be with your kids 100% of the time, not share them with an angry, uncooperative Ex.
I've been in a very similar situation, although my DD was younger at the time. I decided to stay. I got extremely close to leaving. So I'm going to give you some tips.
First, you badly need some friends and you need to insist on seeing them regularly. Tell your DH it is not negotiable. Please, stick to female friends. In your vulnerable marital state it would be extremely easy to fall for a friendly man, and adding an emotional affair to your issues will definitely make it more painful.
Second, you need to take regular exercise. I know this seems an unrelated and odd suggestion, but it will raise your spirits immensely, give you confidence, set a good example to your kids. Again, tell DH this is non-negotiable.
There is no need to plead for this "me-time". You simply state, "DH, I'm struggling with my mental health in this marriage at the moment. We aren't as close and happy as we used to be, or as I hoped we would be. I have thought about what would help fill the gaps, and I concluded what I need is X, Y and Z each week. I am counting on your cooperation with childcare. If this doesnt happen, I foresee that within 12 months I will be very depressed and put marriage will go from bad to worse, which is in no one's best interest. So, now we have sorted that out, please reflect and come back to me if there is anything YOU need , to help you feel happier. I see you get angry and frustrated, and I want to help because it is no fun for me or the kids to live with this. So let me help, if i can."
Second, once you have regular exercise, hobbies, friends and social life in place, you need to confront your DH calmly and honestly about the marriage. Tell him, the lack of sex is a huge problem for you, and you need his help to move past this. If he won't have penetrative sex, ask him to satisfy you in other ways - there are MANY more than one way to please a woman.
As an aside, have you considered a Mirena coil? It might work for you both and doesn't seem to have the side effects of the pill.
Thirdly, be aware that nothing will really fill the empty sadness of having a marriage that lacks real companionship and mutually expressed love. But, if you can reduce your depression, if you can stop his angry outbursts, if you can start to improve aspects of your sex life - it is possible it can be enough.
And, if it is NOT enough, or your DH fails to improve and you fail to feel better about everything, then you might need to consider leaving.
In my situation, i had literally rented a home and furnished it - I told DH I was moving out. It took THAT much of a shock to really get him to address his anger, stop stonewalling me and glaring at me. It wasnt an instant fix but we BOTH compromised, we BOTH changed our behaviour. Now, if he is grumpy, I call it out immediately and he changes his demeanour. I don't let him sulk, I don't let him snap or moan at my kids. But equally I don't insist on having the kind of marriage I'd really like, and I put up with more than I ever thought I would.
So, you can take some of these actions alone to start with, but ultimately you have to have serious words with DH and insist on change. At the very least he must know he cannot get away with treating you badly and giving vent to his anger in your own home.
It will be an extremely tough journey. Tougher than a divorce? Maybe. Who knows? But if you want to give it a try, do so.