I do understand. And yes, it was only chatting which isn't the same as actual contact. Except he lied about it. And now he's essentially checked out of life.
If you want to stay together then get him to a doctor because from my side of the computer screen (I am not a doctor), he seems depressed. And you will need to do the heavy lifting here if he will not (spoiler alert: he won't), if you want the status quo to ever change.
Your children can still be able to see and spend time with the father they love even if your relationship ends or you separate.
Right now, you have 2 children in diapers and one parked in front of the TV, depression, a full-time job, and a side hustle. You cannot keep this up forever. Something's got to give, and I would bet dollars to doughnuts it'll be your health.