I think this is based in a more old-fashioned style of holding a dance, where "going with" a person didn't mean being glued to their side the way it is more commonly done today. IIRC, anytime from about the early sixties on back, it meant they picked you up and took you to the dance, you were expected to dance with a variety of different people, and would dance maybe one or two times during the night with your date, and then your date would take you home. And it was also based on the rule that you have to at least pretend a prior commitment in order to turn down an invitation, and if you showed up at the dance after claiming a prior commitment (like washing your hair!) it would be obvious you'd lied. So the only acceptable excuse was "I'm already going with someone," because that wouldn't be proven a lie when you showed up at the dance. This was also a time when it was shocking to go stag.
These days, and for some decades now (I'm 34 and our dances were like this), going with someone to a dance is pretty much agreeing to be plastered to them all night. It's very much an individual-romantic-date thing in a way it wouldn't have been in, say, the 50s. And it's also not frowned upon to go stag or with friends, in most instances (though I know there are exceptions). So turning down a date is not as big of a deal because a girl does have the right to decide she doesn't want to be romantically involved with someone.
I think Miss Manners is just behind the times and has no idea what dances are like now. "Arrive at the dance with me and dance one song with me" is a very different offer from "slow dance mushed up against my body all night."
ETA: And also in the fifties you were expected to "date" a bunch of different people in a really casual fashion, until you decided to get serious with one of them and go steady, while today people usually go from friends to "in a relationship" with no intermediate step. It's just a very different time, and Martin is stuck in the old one.