That has to be the most horrible thing I've read. It takes a special kind of insane to say that about their own child, not to mention everything else.
I'm so sorry you're here, but you're in the right place. If you haven't already, read through the healing library. There's a wealth of great information in there.
He says he can't live without you and the boys, but what is he doing to show you that? He needs to offer you full transparency, give you access to all devices and accounts without deleting anything. He has to write a no contact letter to his affair partner.
Sending you strength and (((hugs)))
You will get through this.
Post often, it helps.