Hi & Welcome.
I am a WS too, just identifying that since you are new.
First, let me address the name calling and tattoo you wanted:
Right now, you are ashamed of being caught. That is not a judgement, but that's the place we all land in the beginning. It will be months before you get to the remorse stage. But, let me tell you what I learned about being in your stage:
1. The other posters are right, you are and probably will for some time make this about you. I used my shame kind of like a shield early on. I would say things like this (though the tattoo would have been extreme, so thankful no one would do that for you) but what I learned is it shut down conversations that were needed. When I would say these things to my H, it deflected. It made it about me. It wasn't about his pain.
2. You don't mention withdrawal from the AP (feeling like you lost something), and maybe you don't but if you do feel that way, make sure that you do everything to maintain No Contact. Block him on everything. Construct a no contact letter and share it with your husband. Ask him how he would like it delivered. He may want to be the one to send it. No contact includes things that the AP doesn't participate in, that means no looking at his social media or checking in on him through other people. I have seen some come on here and not have withdrawal, and others pining for months on end.
3. Agree to whatever your H wants in terms of separation or contact or otherwise. Make this about his emotional safety and the emotional safety of your children.
4. I don't know what to tell you about the kids or what their ages are. We chose together not to tell ours because they are grown and out of the house. It wasn't to protect me from what they might think or our relationship, but to protect them from the stress of it all. I don't know how or why the kids have come to know, but when things settle down a bit you should get into family therapy with them. They are not mature enough to handle this very adult situation and regardless of your poor decisions they still need a mom and they still love you, there is just a lot of conflicting feelings there.
5. Marriage counseling is not for now. IC is for now. I was in IC for six months before we did any marriage counseling. I needed to become safe enough to work on the marriage with my husband. That meant taking full responsibility, knowing at least the beginning of my whys (I am 9 months out and still discovering more on a daily basis)
6. Read the stuff I think sassy pointed out. Not just friends, helping your spouse heal, etc.
7. Justifications about how you weren't all that attracted to the guy or the sex wasn't good aren't relevant at this point. It's a double edge sword: If he was someone you connected with on a deep level and thought you loved, then the BS feels compared and not special. If you didn't like him that much, the BS feels why would you blow up everything for him. You are best to try and get to a point where you see this person helped you in not only destroying your marriage but more importantly that he helped you destroy your husband emotionally.
8. Go to IC and work on your whys and how you are going to fix them. It's a long process.
9. If you really want your marriage back and he does try to work it out with you then you need to understand that this is a very long road...2-5 years. You really must decide that you are committed to that recovery, and know that you do it without having any control of whether it works.
Stay here, keep processing, keep reading. Have faith that this will work out in some form in another and stay focused on becoming a healthier person.