I'm so sorry that he is so dishonest. I also know exactly how bad that feels. My STBXW is on the phone with OM 2-3 Hrs a day but still denies that it is anything but a "friend". It is like once they start down the road of deceit that they cannot return. Mine lies about almost everything now. Even trivial stuff that does not matter.
I made a deal with myself at the beginning of all of this that I would try as hard as I could to save my M. It was between God and I when I quit. At some point I realized that it was better to move on for me and the kids. I hope and pray that he changes but you have to accept that he may not and you need to let him go at some point.