It is hard to believe, but you are doing almost as much damage to your marriage as he is right now, by continuing to contact him. You cannot control him. You cannot reason with him. You can only feed his addiction.
Have you told him how painful contact is while he is seeing someone else? Have you asked him not to contact you? Have you put it in writing so he can refer back to it? Make it very clear you want your marriage but that there isn't a marriage with the way things are now.
I couldnt' have typed a better post if I prayed for guidance.
This is exactly my point, you BOTH are damaging your marriage when you pick up the phone.
No Contact is SCARY but do-able. You just hire a sitter to watch them and to show them out when your husband arrives... you do NOT need to BE there when he picks them UP or when he drops them OFF...
And any emergency re the children he can send to the sitter.
We dont' know what visitation arrangement you have made exactly at this point, but I suspect it is not conducive to NC, so you may have to change it.
Write you out your commitment statement to him as Whatnow said, and THEN END CONTACT with him.. PERIOD.
I know its hard and counter-intuitive, but it is JUST as HARD for HIM, you've SEEN his desperate NEED to keep in contact with you... when you tak that AWAY he is FORCED to make a CHOICE...
He gets YOU and his family or the OW and her trainwreck of a marriage she's running damage control on right now... what do you think he's giong to choose?