Is there an equivalent one in North American English?

Some people say "nature calls" but it's kind of a joke saying and doesn't have same connotation I think

Urge, urgency, call of nature 

You know, I’m not sure there is. The closest thing I can think of is “sudden urge” or “urgent need” to go to the toilet/bathroom but the latter just implies REALLY needing to go, unexpectedly or not.

The closest thing I can think of is the phrase, "When you gotta go, you gotta go." It tends to be used by other people when someone has made a sudden exit or relieved themselves at an unexpected place/time. It comes up more in movies/TV as a comedy device than in real life.

[Prairie doggin' it](https://youtu.be/fMYspV6jNtk?si=u1M-PdNq_Xz1YinD)

“To be caught short” is also said in the US sometimes, though I’ve only really heard it from people born before like 1970. It’s vague.

Not really. There are plenty of euphemisms and idioms related to needing to pee/going to pee: making a pit stop, draining the lizard, taking a whiz, powder one’s nose, nature calls. But I’m not sure there is one that particularly captures this oblique phrasing and specific scenario. There is “to be caught with your pants down,” but that’s a metaphor for being seen in an embarrassing or compromising situation unprepared.

“To be caught off guard” but n that’s not really specific to going to the bathroom. It can be used for anything unexpected.

"I just sharted"

No, no equivalent. But there should be.

there isn't a common way to say this in my dialect

I have a bad case of diarrhea

Wait until we ask North Americans about the word shoeburyness.