Have to meet the "new man", how to approach this?

STBXW and I are still in the process of getting divorced, were living together full time up until recently, we are now "nesting", aka splitting time at the house, when I'm at the house she goes to the guy that she's been secretly seeing for months (started when we were supposedly working on things), of course she doesn't call it an affair, and thinks everything is totally normal and ok, but whatever. She's talking about wanting to introduce our two young children to this person and wants them to go live where she is. I of course say nothing is happening like that until I meet this person, so I guess I'm going to have to in a couple of days. Those who have been through this, how did you handle it? What kind of questions did you ask and how did the conversation go? Of course I want to fight every urge to just swing on sight, knowing that would be pretty dumb. Any advice is appreciated