Tristan, whatever is wrong with your wife, she is not emotionally healthy. That means it's doubly important for YOU to be emotionally healthy, for your kids' sake. They already are going to have to deal with whatever chaos is going to come from your wife, especially if she does continue in a R with the OM and they have that to deal with.
So, I have to disagree with SmQ..I mean do your kids really need to have an OW to go with an OM? That's disgusting. Are you going to teach them to respect marriage or not? Until you are divorced, they should not see you dating. This is all very confusing to children. Getting together with a group for social interaction is one thing, but what's being suggested is some kind of juvenile "fight fire with fire" attitude that is not the mark of a mature man. Last time I checked, none of us were in high school.
It's nice to be flattered to raise our self-esteem, but trying to raise your testosterone by getting involved with other women is JUVENILE. Self-worth has to come from within, not from flirting. That's only as good as the last encounter. Also, you could end up, in the state you are in now, falling for someone, and wouldn't THAT be a fine kettle of fish for you to be in. You have enough problems with your family.
Work on you, go to your IC, build yourself. Don't play games. Games are for emotionally UNhealthy people. Get healthy. Look at why you were attracted to someone like your wife in the first place. Not just everyone is attracted to borderline type women. Find out why you are, ESPECIALLY before you spend significant time with other women. What is it about YOU that is drawn to someone like that.
Your children need you to be rock-solid right now. I know it's hard, and it sucks, but it's the truth. If you want to be a "real man" then be a great dad.