Hi Sansone,
Sorry to have you here, but now that you are, you're in a good place for healing and hope.
First step is accountability. It sounds like you're starting with IC, and that's good. It looks like you've struggled with drinking too. Getting into AA and getting a sponsor will be a great help- it will surround you with people looking to change their life for the better. Looking to gain accountability and grow. Once you have healed, it may give you future opportunity to give back in the form of sponsorship for yourself. You need people around you who are on this journey of self-discovery, growth and change to encourage you in your progress. It's enormously hard to do it alone. So don't do it alone.
I see that one of these men definitely has intentions, and it's not innocent by any means. I feel I should be trying t win her back, I should be trying to prove to her that in the future I would be a good husband. She doesn't need to be out in the world getting fucked by other men. She isn't that kind of woman.
What Bulcy said, THIS is misogynistic. You're telling her what she needs. How do you know this man has bad intentions for your wife? What defines a bad intention? Desire for a casual encounter? Desire for a relationship with her? None of those are bad in my book. None of those are any of your business any more. She is not your wife. She is only the mother of your child/children. That is the only way you get to look at her from now on. Each time you find yourself with these thoughts, change the channel. Say to yourself, "She is not my wife anymore. I do not get any say in her personal life." Then, find something else to do. Read here, watch the news, work out, take a walk.
Get a dog. Get a cat and play with it. All healthy forms of distraction. You need to rewire your brain to change your thinking. Pets are unconditionally loving creatures and their distraction, attention and the care you give them will help reroute your brain into healthier places than obsession over your ex.
Coming around to the paternity question, I would take the advice here of others and work with a lawyer to have it done. You say she's struggling with money and that your choices led her to that place. Establishing paternity would increase the support you give her. Do the right thing, demonstrate your ability to be honorable and trustworthy as a co-parent by arranging the paternity and being as generous with child support and spousal support as you reasonably can. There is no reason she should be working a night shift to provide for the children. You can make this happen for her by providing her with money to care for the bare necessities- food, rent and car payment. You can even see about going over and beyond by paying daycare, healthcare and all childcare (clothes, toys, furniture even) expenses. You can set up an HSA account where she can submit expense reimbursement for their medicine and doctor visits.
So, in review:
-Get into AA and find a sponsor
-Change the channel on intrusive thoughts of your ex and what YOU think is best for her
-Look at getting a pet, it will ease the loneliness and give you positive companionship
-Get the paternity test done
-Look at what the maximum support you can give to your ex is AND DO IT WITHOUT BEING ASKED.
All this will help YOU be able to look at YOURSELF and see a better person emerging from the ashes of the old broken one. It will help YOU gain your SELF TRUST AND ESTEEM back for yourself.
Your exW is no longer there to mirror you back to yourself in all your glory. It's time to face yourself Dorian Grey style and see yourself as you really are. Start with the person you are today, find out how you became this man who cheats and minimizes and allowed your wife to be betrayed in the most painful manner possible. THEN you can look forward to who YOU want to become. Hopefully it's a man YOU can respect and trust.