Your sentence should read:
Despite knowing full well the negative consequences of our impatience, why are (so) many of us
In a question, the "are" changes its position. Also, the use of "so" in this kind of expression, while not necessary, is common and natural.
As for "know full well," it is true that
1 "full" is a 'flat adverb,' one that does not end in -ly,
but it is also true that
2 "know full well" is one of the most common collocations that uses this form. This can be seen by how many dictionaries give this expression as an example for "full" as an adverb. See the examples in thefreedictonary.
In my experience, know full well is more of a spoken expression, whereas "know fully well" is better for formal contexts, including much writing.
In your sentence, I would use "know fully well," because the flat-adverb version sounds dialectal, slangy, or just too informal for that sentence.