Is THAT what that idiom means? I knew it was said on exiting a room, I never realized it was more specific than that.

In the uk I’ve heard “I’m going to see a man about a dog”. (she went smoking)

That’s another one with a British origin (originally “see a man about a dog”). Still widely used in the UK to refer to leaving for any unspecified reason, although this is often to go to the toilet. (Also was used in the US during prohibition to refer to going for a drink.)

I also heard a more crass one: dropping the kids off at the pool