And "oughtn't", like "mustn't" is much rarer in American English. Quite honestly, I can't recall ever having used or heard "oughtn't" used in the States in my life
I think (no data to support, just an impression) that "ought" is more common in the South than other parts of the US, for me "oughtn't" is a little folksy / old-fashioned sounding but not as jarring and wrong-sounding as "mustn't" (which does sound wrong in US English, like 'shan't')
I have the idea (again no data to back it up) that for USans that use 'ought' it carries a stronger hint of "but I probably won't" than should does.
"I really should go to the doctor's" (I don't want to, but I probably will)
"I really oughta go to the doctor." (but I probably won't unless forced).
(don't ask about the 's - my intuition was to put it in with should and leave it off with oughta but I don't know why)