Hi Exrev, my story is very similar to yours. We are 3+ years out from my husband's A. He has been a liar his whole life. I actually considered D before his A because of all the lying. Then I decided they were mostly innocuous lies, and I could live with that. I thought I always knew when he was lying.
In retrospect, I/we realize that the lying was the slippery slope into his A. He kept reassuring me that his "friendship" was innocent. That really killed a part of me, I believe.
He eventually ended up openly answering my questions about the A after one week of me dragging information out of him. His guilt was eating at him. But then he lied to be about being no contact with the filthy whore for 3 months after. In his mind, if there was no sex, it was okay.
I made the condition for R total honesty. But like others have said, counselors have told me that he may not be capable of that, since this terrible "coping mechanism" started in his earliest childhood. As a result, I have PTSD, frequent nightmares and terrible self-esteem issues. I work out and get attention from men, but it is fleeting. I have come to realize I will always feel like damaged goods.
So we limp forward. He insists he is telling the truth. And when I think he is lying I tell him, instead of choking it back and accepting that is who he is. It is unacceptable to be lied to, even about small, seemingly trivial things.
I will pray for you and your sweet children. It is certainly possible for you to provide them a safe and stable environment even as a single father. I hope you find your answers. Take care.