Though it's the most difficult with black speech, the problem extends beyond African American dialects. The Local Color writers like George Washington Harris would also transmogrify Appalachian and Southern dialects in order to make the speakers pleasantly clownish and peculiar. In simply reprinting you fall into the trap of seemingly endorsing the caricatures.

However, it seems to me that you don't want to entirely lose the original speech. In Tennessee ( and likely elsewhere in the South) there's the term Buzzard Luck, which can be defined as "when you can't kill anything and nothing will die". But when I first heard it, it was "when you can't kill nothing and won't nothing die". I defy anyone to tell me the first version is better! Eudora Welty managed to write authentic southern voices but do it with standard words. Your version of your quote seems to do that.

But you've also got to take notice of what your source says verbatim, whether it makes you wince or not. I would think pretty safe thing to do would be treat it like many people treat a quote translated from another language: put the translation in clear English in your text, and put ( or, I guess we could say bury) the quote in a footnote, accompanied by " the original: ..." I'm not sure but what you could also use [sic] , to show that you're just copying it, mistakes and all.