Rain, lots of good advice here. So, as we are saying to you. Come here first and let it go here. Write all your frustrations, desires, pain, etc.
It also helps if you read what you are writing. It makes you think and find out what is going on inside of you.
I also think you do not need to blame yourself. Be gentle to yourself now, this is a situation you have been enduring for a long time, so the stress tends to build up very fast.
Remember you have the little ones and most important, right now they have only you. Don't be mistaken thinking that they do not feel all the energy around them. They may not know the extent of the problem, but dad is not home and mom is super sad.
And again, it is very hard to just accomplish all the above, be strong and super mom. You will make mistakes once in awhile, you will regret this or that word that came out without thinking.
Situations emerge and you do not know how to deal with them at that exactly moment, but don't bit yourself down, it is part of learning, growing as a person you want to be.
Question: Do you have any family or friends that can help you with the kids, so sometimes you can go out just to breath by yourself?
Calm, breath, let go of the anger, look at all this like your marriage got a bad cold and now it needs to go light to get it feeling better.
You will do it, an affair may happen for a reason, but it can be repaired and get stronger then ever, because then it will be a choice.