You've got quite a few red flags popping up in your posts OP
It's like he just can't stop, and he can't put their feelings ahead of his own. There have been a couple of times when I've been really upset and that has made my daughter really upset too. I've asked him to please stop doing it because of the harm it's doing to our children but I don't see any progress
You've asked him to be a mature adult and hold off on continuing a lecture argument until your young DC's are in bed but he can't see past his own selfish need to talk at you, berate you and point out all your flaws (another red flag there, your'e the one with the problems not him, it's all your fault due to your multitude of flaws - I must point out its only him seeing and telling you about these so called flaws so consider the source before taking then to heart) . When he disregards yours and your DC's feelings like this he's being selfish and abusive.
I've learnt to avoid most things by just not saying anything or keeping my upset to myself until I'm alone
So you're walking on eggss around him, modifying your behaviour to avoid him lecturing you and upsetting you and the children. This is no way to live. It's a classic sign you're living with an abusive man.
My 6 yo daughter is always more clingy after this happens, and that makes him worse. I've tried to explain that that's normal but he thinks we're ganging up on him.
Are you serious?! He thinks a 6 year old is ganging up on him??? 
🚩🚩 🚩 what a petulant man-child he is.
Do Not have councilling with this man, its never recommended to have councilling with an abuser, they will use it against you and the abuse will get worse.
Think long and hard about what your children are learning about relationships from how he is treating you. You don't need to live like this.