I have decided to file for divorce on grounds of adultery but as its currently stands will not be naming the OW. This is not an easy choice for me to make however I do not think it will make me feel any better because apart from the courts, me Husband and OW no one will know.
Those that matter, his family and hers, will know, no matrer how you file. Even your kids, eventually. Their workmates will know too, as they don't seem inclined to make even a pretense of not to have begun before the divorce. It will effect the career of at least one of them, as once they are known as a couple, they cannot be on the same path promotions wise.
Stop reacting to him, put your preferences for timeline regarding the kids and OW out there. If he is moving to his own place, there is no reason at all to hurry her into their lives. The children need some months to adjust to the new normal of two homes before being asked to accept a new partner. It is different for OW's child as the huge change of splitting the family and mum and dad having different homes and friends is already a fact for that child. As it will be for your children, someday, when you have someone they need to meet.
It isnt about not allowing them to meet her, it is about regulating the changes to a pace they can adjust to. Research supports going slow. But while this allows you to control and delay them meeting her, you cannot prevent it altogether. Unless you are lucky enough their new relationship doesnt survive his actual availability. Barring that, she is part of their normal.
Discuss with him the official explanation for why daddy isnt here anymore, and why Mummy doesnt come to Daddy's house with us. It needs to be simple, and used by both of you. They are so little, and really only know what you tell them, so it needs consistency. 'Because daddy and mummy each want their own house now.' is probably actually quite enough for such small children. Mummy and Daddy having their own homes can be as reasonable as any other family members having their own homes, if you and all the family they know present it that way. They havent a clue about love and marriage and divorce and division of property. Mum lives here, dad lives there, grandparents live here, Friends live there. As with my younger dd, seperate parental homes will be all they ever remember, so just be factual.
As they experience two different homes, and the grandparents we see with dad vs the grandparents we see with mum, and the different neighbors, eventually, the different friends, incl. OW, will simply be facts of their lives. They are unlikely to remember ever living in one house, and in the end, may not remember not having a stepmother. My younger DD does not remember not knowing her stepmother, though she remembers the wedding and moving in.) It is not the life you expected to give them. But it is the life they are getting, and they will accept it more easily if you do.