Quotes: Change them up or leave them alone?

I believe a quote should never be changed.  If a direct quote had problematic features, the easiest solution would be to summarize the information the source had given.  That way you still have the facts and maintain a high level of credibility in your article.

For example quoting exactly what he said:  "Thanks to my opponent's dirty campaign, I ain't never gonna run again," the mayor said, a reporter may write: Citing his opponent's "dirty campaign," the mayor said he wouldn't run again.