How old was your H when his birth mother left him? If he was an infant, she may not have what we think of as completely healthy or even normal motherly feelings for him. Her H has dumped her and she is texting her son how many times a day? She hates his W and convinces him to move in with her (who he doesn't really even know) instead of working on his M? Look, I had challenges with my MIL, but the more I read about this case, the more it sounds a bit abnormal. I don't mean I think they are doing something incestual........yet. However, if she has had no contact with him all these years......then it's possible she doesn't have that motherly bond, that we have with a child we raise. I can't remember the clinical word for it, but it happens when birth parents and the child, or siblings, have not lived together and/or had contact since birth. Then when they meet many years down the road, they don't have that bond that is naturally formed between family members.
I remember a similar case some years ago, except that was a daughter who went to live with her father, who had not seen or contacted her since her birth. (In fact, I know of two cases where this happened.) Sadly, it resulted in a case of incest b/c he was a man who saw her as just another woman. The daughter was confused about their emotional relationship......as one might only imagine when never experiencing a father's normal affection/love.
Your H's birth mother could be turning to him to fill her emotional pain, just like any woman might turn to some other man when she has been rejected and left alone. I could see all sorts of emotional unhealthy outcomes, when both of them seem wounded and immature to deal with this type of situation. He definitely sounds confused, and maybe a bit brainwashed by his mother.
This is merely speculation on my part. I contemplated whether or not to mention it, b/c I know something of this nature surely won't help your feelings. I only wanted to warn you, b/c it does happen sometimes, and if there is a slightest hint the woman is unstable, it will take a lot of strength and emotional maturity for your H to pull away from her. Ask yourself why he feels he has to choose between his "new" family and the ones who have been there for him all these years. I mean, I've heard of MIL's from hell, but there is something that definitely feels very unhealthy in all of this. Since meeting his birth mother, he seems to have become weaker, signs of immaturity, and dependent on her. She has become his crutch, and maybe vise versa.
Anyway, you might think about discussing it with your IC. In the meantime, I would consider pulling back very hard and very quickly. You can't help your H decide what to do or who he wants more. He has to figure it out for himself, and it may take him going through some stuff before he comes to a final decision. Not an easy thing for either of you, but you have to protect yourself. When he sees you moving on without him, that will do more to work in favor of your a future with him, than most anything else you can do. (((hugs)))
It is not about what you feel should work in your M. It is about doing the work that gets the right results. Do what works!