It is said that the biblical phrase directly translated as “cover one’s feet” is such a euphemism. Judges 3:24 and 1 Samuel 24:3. I don’t think this was ever in wide currency as a euphemism, except perhaps jocularly.
When my father needed to stop at a gas station during long road trips, he would say “I need to sharpen my skates.” That might be a Midwestern thing as he was born in rural Kansas in 1913. I’ve never heard anyone else say it,
Down South we always used “I got to go see a man about a horse” when telling others we need to go to the restroom.
Nice one. I couldn’t use that where I live right now, though. My landlady actually owns two horses who live on the property.
It’s possible it was a euphemism in this occasion, but “taking a constitutional” usually means going for a healthy walk.
Indeed, some people go for a nice walk in the morning—not a myth!