Quote: he's not agreeable to having a part in the baby's life. He said, "Our son is a part of me and you. I don't want to have anything to do with OW at all. I don't ever want to have to see her again."
I asked if he's sure of that (mainly because I disagree with his position, but I didn't tell him that, of course). He said he would be willing to pay child support if that's what he had to do, but he still wouldn't want to be part of the baby's life because the baby -- and having to see her mother -- would serve as a constant reminder of what he had done to me and our son.
I know some would consider this politically incorrect, but I happen to agree with your H.
IF (big if) OW is really pregnant, the very best thing for that child would be to be given up for adoption to a 2-parent home that was unrelated to this mess in any way. Second best would be if OW meets and marries a new man who raises that child as his own.
I DON'T think it is good for that child, or your kids either, to be raised with the knowledge that it was conceived from an affair and feel somehow "second best" to your own kids, which would seem inevitable.
I think H should meet his financial responsibilities. I don't see the point in getting an attorney until OW sues for child support (again, she hasn't even had this baby yet, and I for one still doubt she's even pregnant). At that time you can request a paternity test.
It's possible she will have the child and not sue for support because she doesn't want to share custody. If that happens, I would let it lie, and just set up a special savings account to set aside money for this child (in case she sues for child support later, or it could be given to the child for college when they are grown.)
