Totally can not believe this. This goes way beyond forgiveness and your trust for your husband and everything else. You need to get her out. Simple.... Say look, she needs to leave yesterday. I will keep her child until she gets on her feet, but she needs to find a new man for herself. If he puts up a fight tells you you are unreasonable you kick him out as well. This is not healthy for you, your children, his children, her children, your husband and possibly not even the neighbors. Maybe for her, but too bad she screwed up years ago.
Listen, she has already showed no respect for marriage, she conceived another mans kid during marriage. HE obviously wishes it never happened and somewhat forgave her since he adopted her son. I understand he was not at fault, but wow!!! I assume he will have to pay child support do to the adoption. He has some sort of heart, I can not explain, but she definitely has a hold on him.
Lose her now. Be very firm with him bc this has potential for reconciliation and disaster. Normally I would say that reconciliation is good but you have kids and to me my kids come b4 someone elses or anyone else.
Hope, you can pull it together, I feel sorry for you.