I don't think you handled this situation badly; in fact, I think you used quite a bit of restraint. My favorite part was when you didn't waste the effort of calling him out on his lying; you just rolled your eyes at him.
But as you seem to understand, engaging with him at all is a waste of time and does nothing but get you worked up and upset. So here's some practical advice:
Is all of his stuff out the house yet? If not, I would pack it all up, leave it outside, and give him specific days/times to pick everything up when you're not there. If that's not a possibility, then make it a point to leave whenever he comes to the house. Go for a drive, go for a walk, take the kids to the playground, whatever you need to do.
You need to limit your interactions as much as possible and when he does try to engage you (by saying stuff like, "I'll help you out around the house") respond to him as if you were an apathetic teenager being asked boring questions by a distant relative.
"What's new at school?"
"Nothing." <goes back to texting>
I think the upcoming months are when things are going to get challenging. From what you described, it's clear that it really hasn't sunken in for him yet that the marriage is over. I really believe that he thinks he's going to be able to tramp around for a while like a stray cat, and you'll leave the door open for him when he wants to come back. Mark my words... he's going to try to delay the divorce process and drag it out for as long as he can, especially if OW is pushing for marriage.
You really need to prepare yourself for when he comes crawling back and starts promising you the moon and the stars for the chance to fix things. He might say "I was such an idiot I didn't realize what I was losing, I'll be the best husband ever!" What that actually means is: "This wasn't as much fun as I thought it was going to be and I don't want to lose half my shit." If you end up taking him back, you're going to end up spending the foreseeable future feeling like Plan B, especially after his neglect and emotional abuse of your children (and yes, telling your child--especially one who is sick-- that you're going to abandon the family is abuse and it's unimaginably cruel).
BW, 40s
Divorced WH in 2015; now happily remarried
I edit my comments a lot for spelling, grammar, typos, etc.