using too many words
Prolix is used correctly. Let's break this sentence down to see why, starting from the end and working to the beginning.
- "[U]nnecessary flourish" means there is a large amount of words that aren't needed in order to understand what is being said.
- If the author resists the temptation to use unnecessary flourish, the writing is simple and concise--it is not prolix.
- Since the author does not resist this temptation, we take the inverse of the above
statement: the writing is prolix.
Is the second part of the sentence a double negative?
Unless you consider resist a negative word (meaning using it changes the outcome of the rest of the sentence), the second part is not a double negative. The only negation here is not.
Often times it is very helpful to figure out negation starting from the end of the sentence because negative words change what comes after them.