The other women has very little to do with this. Your husband has made a lot of little choices to get where he is and he isn't easily coming back.
One of the things that MIGHT help you is actually working to MOVE ON and have him served with divorce papers. He is NOT in a great position post divorce as even though you probably will have to get a job if you haven't already got one - he will still be paying you child support and spousal support and you should absolutely take what you're entitled too - since he's entitled to his happiness - you're entitled to support. Period.
If he REALLY wants to work on it then he will go with you to a therapist who specializes in infidelity. He will realize he needs to re-earn your trust back by opening his life back up and literally giving you the keys to the business and perhaps making you an equal partner in it if you aren't already.
Check out this book as it's super helpful in understanding what is required to recover in situations like this:
Healing from infidelity regardless of degree can be very hard.
The Wayward spouse really needs to recognize what led them to make all the little choices they made along the way. Usually individual therapy for both parties is best prior to couples therapy and that couples therapist (as well as the individual therapists) should have some training in dealing with infidelity. It's a very challenging subject and sometimes a well meaning 'non-judgemental' therapist approach can actually ENABLE really really bad behaviors.
It isn't that the wayward spouse needs to be judged but their behavior needs to not be condoned and reinforced.
I really found this book helpful:
If your husband isn't willing to seek qualified individual therapy AND isn't willing to read the book and own up to their part of this situation then you should not stay with them - they will just do it again - what you're seeing today is an act.
So you recognize what potentially is your part in how she felt but please know that it is not YOUR FAULT that she made the decisions she made to start an affair - emotional or physical.
It isn't' your fault.
They weren't your choices, they were hers and it wasn't one choice. It was a lot of little ones probably starting with not communicating with you about how she felt.
Maybe you can come back from this and maybe you can't. She should go no contact. You should both get separate individual therapists who have training in infidelity to help you though this and then also find a separate couples therapist who is also trained in infidelity to help you come back together as a couple. There is a subreddit that can help you called r/AsOneAfterInfidelity - it can help. I also recommend the following book for both of you. If she will not read it then file for divorce. PERIOD.
The book is worth the read. It has helped a lot of folks I know through times like this.
Don't do the pick me dance though - tell her she has to end it and go no contact and provide you access to everything or you're done. Then BE DONE and actually file for divorce. It's the best thing you can do to get her out of her affair fog and bring her back to reality.
If she's a stay at home mom, have her go back to work - period. Don't divorce her yet - get her back into a job, sit tight and get that squared away and THEN file for divorce once she's working again so that she's able to support herself and you can reduce your support obligations.
As a Betrayed Spouse - I found this book to be very helpful in figuring out what I needed from my wayward spouse to figure out if we could make it or not:
https://smile.amazon.com/gp/product/006304529X/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1
It's called 'After the Affair' by Janis Spring.
It was really helpful in showing a bit of both sides, you made a lot of little choices to do what you were doing. Your willingness to look at yourself is really important and I commend you for putting yourself out there and being willing to take responsibility. That's a big deal, you have to take responsibility for your part and honestly you will bear the largest part of this burden it will seem but the burden that the betrayed bears is one you cannot see and one you will have a very hard time relating to and understanding.
You've already mentioned one of the key things - the smallest thing can create storm of emotions within the betrayed spouse and then you're on the hook for being the target of all that, anger, sadness, etc.
If you really want to recover - you're going to have to accept that role. Your partner has been extremely hurt by this betrayal and the symptoms are often likened to PTSD type symptoms. I was actually diagnosed with PTSD.
My wayward spouse could not even consider that she was wrong. She wanted to actually keep us both! It didn't take me long to decide that divorce was the only acceptable answer for me but recovery is possible and the relationship afterwards could be (I hear) much better going forward.
Your wife has a responsibility as well, this book helped me understand my part. I did have a responsibility for how she felt in our relationship. I did not have a responsibility for her choices. She could have chose to communicate with me better or just, honestly, divorce me before she started another relationship. I'm not responsible for those choices.
Good luck. It's tough being the 'villain.'
If your husband isn't committed to earning your trust back and acknowledging his responsibility for the situation then the best recovery option is without him.
This book details it really simply and I loved it: https://smile.amazon.com/gp/product/006304529X/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1
Basically it outlines what both parties have to do to get to reconciliation. If he's not doing his part of the work then it's just not going to get there without you compromising yourself in ways you likely aren't going to be able to live with.
If you really want to change find a therapist who specializes in sexual addiction as they are usually also capable of handling issues around infidelity. YOU made these choices to step out of your marriage. YOU are responsible for them.
Seems like you might want to recover from it and maybe you can. Often though, you can't.
I like this book for both sides of recovering from an affair:
After the Affair, Third Edition: Healing the Pain and Rebuilding Trust When a Partner Has Been Unfaithful Paperback – August 25, 2020 by Janis A. Spring (Author)
https://smile.amazon.com/gp/product/006304529X/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1
If you're serious about staying together then get her a copy as well but dude, you messed up.
You're not alone. It takes time and a lot of work on yourself honestly. I found this book to be extremely helpful: https://smile.amazon.com/gp/product/006304529X/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1
It also will depend on the state your in and if it's a no fault state or if it's a fault state. Your affair in a fault state could have significant impact on the asset division. Having been cheated on myself, I hope she is able to get what she is entitled to from you. I live and divorced in a no fault state and now pay THOUSANDS monthly to my lying, cheating, manipulating, abusive ex-wife.
My advice to you, for your children's sake is to worry less about the money - it will work out the way it is supposed to work out but do not EVER use your children as tools. Sometimes it's hard not to do unintentionally even but it's a terrible thing when a child is turned against a parent (my oldest).
Apologize to your soon to be ex-wife. Own it, become the man you couldn't be when you made all the little choices you made that took you down the road to cheating.
Read this book: https://smile.amazon.com/gp/product/006304529X/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1
It'll help you understand her perspective a bit more.
I'm sure she is in terrible pain right now. You might be too. Your children's world is GONE.
She may bear some responsibility for how you felt when you stepped out but she isn't responsible for your choices. You are.
You should get 50/50 with your kids. Your STBX will have to go back to work earlier than planned and childcare expenses will rise quite a bit (especially with twins).
Your husband made a choice to engage with someone else to get his needs met.
That isn't your fault.
You might have some responsibility for the feelings he had but NOT how he acted upon them.
Your marriage counselor is out of her god damned mind. I had one like this and the fact is this kind of approach essentially means they are not trained to deal with Infidelity issues.
Get a new therapist who IS trained to deal with issues around Infidelity.
Both of you should read the book: After the Affair (https://smile.amazon.com/gp/product/006304529X/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1)
If your Husband refuses to read the book in a timely fashion - it is short - shouldn't take more than a week if it is important to him then you should file for divorce. He's past reconciling likely but the only thing that is going to show him that you're serious about your part is to actually be serious. File for divorce, he is likely in the fog of his affair and it's going to be real hard to get him out of that and get him back. It may not even possible.