Do NOT feel sorry for him. He has done this.
Women are brainwashed into caring and nurturing and empathising and responding to puppy dog eyes, sob stories and fixer-uppers. He's her problem now - tag, she's It.
Search the house - Is there any more of his crap that can be bundled up and dumped at hers? The more the better. She'll hate it.
Quite right that EVERYONE should know what he's done. This is you taking control. He does not get to hide, lie and conceal any more. Be aware he may try to take the narrative, sugarcoat and spin a story about you - get in first, with the clear truth.
Tell your children that daddy has decided he wants to live somewhere else now.
Be careful - Arsehole though he is, he is still their father, and as long as there is no abuse towards them, facilitating a relationship is wise for everyone's sake.
Preventing him seeing them or being awkward or badmouthing him can be taken as parental alienation, and the courts can take a dim view of that.
But be cool, together and businesslike from now on, facilitating reasonable contact, while setting and maintaining your clear boundaries.
Your home is not available for him to walk in and out at will. I hope he doesn't have keys - he doesn't live there any more. Take him off the council tax and electoral roll etc.
You are not making your time available to him whenever suits him. Set times when he may arrive, and when he must leave.
Do not always be in or available when he wants you to be - take the children out, make plans. 'No. Not tonight, I have friends/family/colleagues coming over' and he no longer fits.
He used to have you and the children in a convenient mental box. He must fit in his box, now.
Certain places are now out of bounds - he does not get to walk into your bedroom, for example.
Do not cook for him or do any other domestic service, as a matter of course. He must ask if it's ok to make coffee, have a biscuit.
This is an option, consider it a strategy: start talking reasonably in terms of 50/50. As in, he is equally responsible for HIS children and their lives. As in, they will be spending 50% of their time at hers. She will HATE that. And he will then start reducing the time he has them, under her pressure, leaving him stuck with her kids ... and missing his own.
Let him suffer. I have zero empathy for these parachuting men, or the vile women waiting for them to land.
And take that legal / financial / child maintenance advice. He's not supporting you, you're an adult who doesn't need that - but ensuring he pays every penny of his duty to his children.
Future prediction: And be prepared for the fake Mr Nice phase - it happens. 'I'm so sorry, I've made a terrible mistake, the grass wasn't greener on the other side, I miss you all, please don't break up the family, can we try again, let's go to bed', promises, tears, presents, extravagant gestures ... All guff. All smokescreens and mirages, fake sparkles designed to prod your buttons, hoover you back in, forgive, enable, facilitate him getting back in all comfy ... ready to DO IT AGAIN because you showed him he could get away with it.
Hope is vulnerability - and they can smell it.