Scotland recently re-wrote the rules on how parents should continue to parent after separating / divorcing. To put it simply: co-parent as amicably as possible, keep things as stable as possible for the kids, keep routines as much as possible. If either of you want to divorce, then the house should be sold or one should buy the other out. If DH contributes equally to bringing the kids up at the moment, then the kids are entitled to see him 50% of the time after you separate. If either you or DH is currently primary parent, then that should continue - no matter who is to blame for the relationship breaking down, and no matter what gender stereotypes say about who should be primary parent. Keep things simple and clear, change as little as possible and avoid any possible conflicts down the road.
The philosophy is that neither parent has an automatic right to either the matrimonial home or the job of primary carer. Instead, the kids have a right to access to both parents, and a right to expect that both parents will do what is needed to maintain stability and create an amicable parenting relationship post-separation. The worst thing that can happen from the children's point of view is that parents get into the a vicious blame game, fight endlessly about who is allowed to live where, or pull the rug from under them by suddenly trying to change what is their normal parenting arrangement.
In your case - by putting your DH in such a horrible (and seemingly endless) living situation (for him), you risk making that amicable parenting relationship impossible. Yes, you have a right to stay in the house, and if you are primary parent it may be, in theory, better for the kids for you to stay and for him to go. However, we live in a messy world and legalities and hypothetical scenarios do not always work. Some parents stay in unhappy relationships because they feel it's best for the kids, or the best way for them to be in their kids lives. You have decided not to live that life. Some parents manage to navigate a gradual, smooth separation where both can stay in the family home until alternative living arrangements have been settled. Unfortunately, it doesn't look like that will be possible in your case.
The next phase is very important. By attempting to take ownership of your children and the family home, you may drive DH out of your kid's lives and your kids will suffer as a result. Your kids do not belong to you. You belong to them, just as DH belongs to them. Unless DH was abusive, the fault for this situation lies at your door - not that you're separating, but in the unnecessarily traumatic way that you are separating. If you want to have a new relationship, you need to take the pain and do that in a way that minimises the impact on DH. Let's not forget, you could stop seeing new OM until you have properly separated if you want to make this whole process easier on the kids.
So, what you and DH need to do is sit down and work out how you can separate with the minimum of disruption to the kids. How do you currently split childcare responsibilities? Who stays in the family home for now should be decided on that basis. If DH is main carer or the split is 50/50 I think he should stay and you should leave as soon as you can. However, if your kids are used to you being primary carer, then that is what should continue, and you should stay. DH doesn't get to upend the DC's lives just because he's upset by your affairs.
Whilst doing all this deciding and negotiating you should remember one thing: one day your kids will want to know why you split up. Do you want them to know that you cheated on DH twice and then forced him to move out? Or caused him to have a psychotic break because you were living at home and seeing OM? In this day and age, with a majority of parents separating before the children have left home, the best any of us can do is show to our kids that we handled the separation with as much calmness, kindness and consideration to them and DH / DW as is possible. Kids are not stupid, but they are not emotionally mature either - they can and do allocate blame, and you need to be able to walk away from this car crash situation with as much integrity as possible.