I know you now this but no matter what happened in your marriage there is no excuse or "good reason" or justification for an affair. Of course your H needs to blame somebody for his crass and disgusting actions to avoid looking deep within himself.
It sounds like it is time to set some hard boundaries. If he wants to split the finances then I would say at this point go right ahead. This is for your self protection. Do not allow him to nickel and dime you for daily purchases or household needs. Simply create a financial plan and that is that. If he doesn't like it then move up the "chain of command" and find yourself an attny.
Everything your H accused you of (ex: squandering money) are things you should be concerned that he is doing. Your H certainly is not unique in his deflection tactics.
The more you question him about OW the more he will lie. Stay silent and then he might (or might not) begin to squirm. While it nearly killed me I have to say that was the BEST thing I did. I confronted H (had hard evidence) then never mentioned it again. Of course he lied but he did live with a very large dose of paranoia and me going dark was far more effective. Granted, I had legal reasons for doing things how I did but still, the emotional stress was present.
I would go as far as to suggest you tell your H you will not be in a three person marriage and you have decided it is time for you and the children to move forward. Go talk to an attny and see what your best options are. Go dark and silent on him unless it is about the children.
On the emotional side I know how painful this is. While there are logistics that need to be dealt with the heartbreak is not lost on me. What else can we do to help you?
