I just want to reinforce and expand on some things that Bucket said:
Your husband is a sex addict. He is not capable of choosing to stop. Let me say that again: he is not capable of choosing to stop.
At least not on his own. I imagine he has periods of abstinence and periods where he says to himself that everything is changing now, but he has longer periods where he feels completely powerless to change and when he falls back on his drug of choice. It is the way he deals with pain. Whenever thoughts of how much he has hurt you and your children come to his mind, it hurts him deeply, and it makes him want to do experience sexual pleasure, because that will make the pain go away.
Whatever else you feel about this man, he deserves your pity. Of course, he has played a huge role in putting himself in this mess. But it probably started with him being hurt somehow by someone else. There is probably a childhood pain that put him into this spiral somehow. God can heal that pain, and God can lift him out of this mire. You can’t. So that means you’re pretty powerless too.
Over him, at least. But God’s attention is firmly fixed on you, and He wants to see you through this. Even though your husband has forsaken you, God will not forsake you.
You made a comment about not having a problem with him using pornography. But you need to see that this is where porn leads. It makes a person curious. It breaks down barriers. Sight and sound is not enough; it makes you want touch. Pornography is completely and utterly worthless and harmful. I know. My own life has been torn apart by it in a thousand ways.
A word about your shame, too: you have nothing to be ashamed of. Nothing. The shame is a lie from the devil. Bring your shame to your Father in Heaven, and He will free you from it – not all at once, but over time.
I don’t know what to recommend you do in the short term, but I wouldn’t recommend a hasty divorce. A hasty separation sounds reasonable – though I would recommend you try to talk to him about his powerlessness and addiction first. He just might open up, if you come to him with a willingness to listen and accept him where he is.