From a mother's perspective, I know all too well that sickening feeling of discovering the 'truth' from a lying husband in the midst of an affair and having all those little kids around to have to take care of. All you want to do is curl up in a ball and sob and wail at the injustice of it all. But then who would take care of the kids? It is absolutely one of the hardest things I have ever gone through. It is SO UNFAIR.
Do you have family or friends around? I know that on this board and DBing in general suggests keeping fairly tight lips on what you are going through, but I needed the support of friends and family to get through those really hard moments. Someone to watch the kids so you could drive to a parking lot and cry. Or a girlfriend to come over after the kids were asleep and let you sob on her shoulders. At my lowest point, my brother and his family came up to stay with me for a week and they cooked and cleaned and loved on my children so that I could just survive. My depression and sadness was so acute that I went to my dr and told him everything and was given some support and tools to get through the following few months.
Be easy on yourself. You don't have to keep your sh*t together. You just have to get through the next 5 minutes. Your body will go into auto-mode and you will put food on the table for the kids, you will bathe them and put them in bed and then you will have some time alone to process or not. It is OK if you are not the best mother at the moment (I beat myself up over this until someone on this board gave me permission to absolve myself of that guilt). You will get that back, I promise.
And although I am not too far ahead of you on this process, you will survive this, I promise.