This might be a language dependent thing. In my native language, Finnish, I don't think I've ever seen a quoted passage, quote marks intact, edited to grammatically fit in a sentence. A quote is a verbatim reproduction, a holy thing. Rather, you would work your sentence to accommodate for the quote. If you cannot do that, you'll leave out the quotes, and they are your words instead of a quote.

Just because a quote doesn't fit your favourite sentence may not mean that you “shouldn't be using that quote”.