Honestly, it sounds like a standard issue midlife crisis/affair with relapse complicating the whole mess.
Why is he lying? Because he can avoid the consequences of cheating on his wife and family and keep you in a holding pattern at the same time. Since his livelihood is pretty dependent on him being the recovery success story, that's another big motivation.
I'm very sorry for you and your kids...at this point, you maybe should just assume he's lying unless proven otherwise.
Sending you a hug...