As a shortening of "bourgeois", is "bougie" or "bourgie" correct?

From this ngram, both bougie and bourgie are so rare in comparison to bourgeois, that I'd suggest avoid using them and stick to the full word.

Even if they are not "wrong", they're far from idiomatic.

According to Urban Dictionary, Bougie is:

"Aspiring to be a higher class than one is. Derived from bourgeois - meaning middle/upper class, traditionally despised by communists."

Example in Urban Dictionary:

"When my friend Miya wears a blazer or Lucy gets a massage I call them bougie cause we're unemployed college students."

I would say bourgie is more accurate. But that's my opinion, mostly because of the 'r'. Bougie is used more as a slang than anything else, bourgie is only known to and consciously used by people aware of the fact that they're referring to the shortened form of the word bourgeoise. Half of the people who say bougie and actually think of it spelled as bougie are likely not to be aware that it was derived from the word bourgeoise.