I might say something like “Oof, gravity is kicking in, I gotta go have a seat…”
Not pooping, but whenever i have gas, I say I ate a kitten. Not many people get it: https://www.youtube.com/watch?v=-sjRO6BQFMw
Dropping the kids at the pool.
I need to pinch a loaf.
“I need to poo-poo in the pot-tay”
“Gotta use the sandbox”