Okay I have read your situation. I understand how difficult this is for you, especially with two young children. However, you have more than enough evidence here that is showing that your husband is not trustworthy, is not serious about rebuilding trust and has deep seated issues that need to be addressed, which have nothing to do with you.
He could be suffering from a sexual addiction/compulsion. This can get better but only with hard work, dedication and long-term counseling for him. Even in that scenario, there is often a chance for a relapse or another affair. His behavior is quite simply outrageous and over the top.
You need to adjust your thinking in that you have not failed in this marriage! You are simply in a marriage that is not working, due to his continual destructive behavior. So, do NOT beat yourself up if you choose to divorce. In fact, you would fare better emotionally if you did get away from this chaos, as you said you are feeing hopeless, worthless currently. The kids would do best in a household that is tension free and where their primary caretaker (in this case, Mom) feels happy and stable. I can't imagine that this is a good environment for them with all of this going on, and your attention and energy being zapped due to this ongoing situation.
In summary, your husband has not shown in any way that he is serious about changing. His promises are empty and at this point, are not to be trusted. My best advice is to proceed with what will make you feel most at peace, in control of your daily life and future and give you back your self-esteem and feelings of worth. This will only reflect well on your children as well. I hope I have been helpful in guiding you in a good direction. And counseling for yourself, of course, would only be most beneficial. I wish you all the best. Please click ACCEPT so I'm credited for this answer. Feel free to continue the discussion with me, even after clicking ACCEPT. Thank you.