I'm sorry this happened to you. I would recommend getting in to counseling right away, both you and your husband. Either he is truly gay, or he's bi sexual. Either way he did cheat on you. You both need to find out for sure. I understand the concern for your children and wanting to keep the family together. Sometimes a marriage can not be fixed no matter how hard you try. But, I would recommend trying everything you can before making a decision. Maybe a trial separation while you're both in counseling would help? We're here to listen and help if we can.
That's really messed up, please don't stay with him, he's very deceitful!
Most statistics say that children of non-divorced parents do better, now forget what I just said, if your husband is doing this, then you can end the relationship, if you want custody, the courts are stacked for mothers, statistics mean nothing, if you are a good parent, then your kids should be fin, although I recommend you see someone for your deppression, anti-deppressants could help.
It no where says he is a bad father!! But yes I would leave. So he could be happy as well as you being happy. Ur babies will never be happy unless mommy and daddy are