Hello, I'd like to help you with your question.
I am sorry that your husband has hurt you like he did
. Infidelity can be very painful and it changes how you feel about your marriage, your spouse and even yourself. It takes time to work through as well if you and your spouse want to work to save your marriage.
If your husband does want to try counseling, that is a good sign. It is worth talking this out. It may help to change his mind about the marriage and be willing to give it a try again. And the counselor can help him see the damage he is doing to your marriage through his actions, including lying to you, which undermines your ability to trust him.
When you have children, you want to try to keep your marriage together so they benefit as well. It is hard to break your family up and try to cope with all the stress
a separation brings. It is always better to try and make an effort before considering a divorce.
If your husband refuses to leave the home, it makes it hard to know how to act. A lot of it depends on how he is treating you right now. If he is making an effort to be civil, then you can either treat him the same way or try to be nice. But if he is acting in a hostile manner and not acknowledging what he has put you and the children through, then it may be better to either ignore him or try to compromise so you both can at least get along but not have intimate contact. You do not want to be hurt worse and if he is unwilling to see what he has done as harmful, he may take advantage of your kindness and see it as a pass to hurt you and the children again.
I hope this has helped you,