Sorry to here that but you are not defining any boundaries for your wife. She says she wants to keep seeing the other man, her feelings are too strong , soul mates, schmoopies blah blah but wants you as her buddy and to be ok with it!! Plan B
So your either going to let her keep walking all over you having her huge slice of cake and gauging on it right in your face. By the way that will go on for as long as you allow it and/or until she gets bored (she may never) of him which could be years by which time she will have lost all respect for you completely.
Or you can say and commit to im not prepared to be treated like this any longer, you love the other man so much I think you should be with him. Lets move foreword with separating.
She is taking the p!ss out of you at the moment because she can because you are allowing her to. Its super hard I know. I am going through the same sh!t. I have woke up though and realised I am better and deserve more than the scraps my wife was throwing me. If she wants to go....Let her go. You'll be great again with or without her.
Be strong for your kids. Be a great dad and that means not allowing yourself to be walked all over. What are you scared of? Losing her? she's already gone. she went a long time ago, you just didn't realise it. Its the same for everyone on here. They don't just wake up one day and think "oh ill have and affair" she will have been unhappy for a long time but instead of speaking to you (or maybe she did but you didn't listen) she has eventually moved on and now met someone else. Its the same every time. My story is exactly the same. I wasn't there for her, she got too close to someone else next minute they're in love, he's her soul mate, lovey dovey, floaty clouds, fairies the works. Trying to nice her out of the affair will not nor has ever worked.
How to act? Just be normal. Friendly, confident funny upbeat. don't act like a complete aloof idiot or she will just see straight through it. It is going to be super hard.
We all come here wanting to save our marriages but at the moment it seems it is only you who feels like that? she may want to stay in the marriage, the security , family time etc. but on her terms e.g. still seeing OM Don't show disappointment, its just shows you as being needy = pursuing = weak which she will prey on.
