The right thing for the children is to have two parents who love them and have a healthy coparenting relationship rather than a toxic one in which neither side is genuinely content, secure and happy.
Yes, this will be difficult to achieve but it's perfectly possible with hard work from both of you.
You cannot have exactly the same set up as before you split up. It simply won't work. So you need to work out what can change and what can't.
You will need to compromise. I don't think you addressed people's earlier suggestion that you find a room in a house share in the short term in order to get everyone used to not living under the same roof but maintaining consistency for now of the kids living in the home still and you doing the wraparound care. This will of course be tough but as I said, it is absolutely not sustainable for you to live together considering what's happened.
Your focus needs to be on coparenting successfully, not on attempting to paper the cracks for the sake of the existing family unit's set up. Lots of couples split up. Lots of couples have to, after the split, have tricky living arrangements and cut back in order for both parents to have safe places to live and have access to the kids. But people do make it work. They may not have the same amount of disposable income as before, or have the number of hours with the kids they'd like but it's real life. The fact that you meeting a woman, rather than another man, was the catalyst for this change doesn't affect what needs to be done next.
Cut contact with the woman, focus on the kids, on coparenting healthily and on not further damaging the dynamic between you and their dad. If you try to make it work, when you aren't in love with him or sexually attracted to him, the relationship will end anyway, just with even more bad feeling and resentment.
It's shit, yes. It's going to be a shit time in the short term. But that's life, that's reality.
You can do it, lots of people break up and the world keeps turning. I know that sounds trite but it's true. This isn't the end of the world. It isn't the end of you being a family - you always will be, as you share your children. You just won't be a family where both parents live together or are a couple.