Why has Southern US English all but abandoned adverb forms?
Solution 1:
"Real" and "awful" as intensifiers go back a real long time, and are much further widespread than the U.S. South. See this ngram. With respect to these particular terms, I imagine people in the South speak English just as good as anywhere else.
For "ran quick", you might actually have a valid complaint, since Ngrams shows "quickly" is much more common. But one example isn't good evidence of a general trend towards adverb loss.
Solution 2:
For whatever reason, Southern speech has kept many old forms. The use of flat adverbs was once more common than it is now. So it's not a case of Southerners (and others as well) abandoning the -ly but rather a case of preserving the non-ly form!
"Old John of Gaunt is grievous sick, my lord" — King Richard II, Act I, Sc. IV.
"the weather was so violent hot ... the weather being excessive hot ... extreme hot ... the sea went dreadful high. - Daniel Defoe, Robinson Crusoe
Common uses of flat adverbs:
Stay close to me.
Drive slow.
Think different.
Hang tough.
Do right.
pitch black
mighty fine
I sure hope ...
Drive safe.
The batter drove the ball the deep.
He tried to go deep with that pass.
Quick, take the shot!
Solution 3:
"Good" as adverb is widespread even if it is seen as an error in many places. I've come across it countless times and I'm not in the South. Also, it's arguable that the "-ly" suffix is redundant since the word order usually lets you infer that these words are adverbs and not adjectives. So just as my daughter, who is still learning to talk, naturally decides to use adjective words as adverbs, it seems natural that this sort of thing would be common. My intuition tells me that this sort of speech is associated with people/regions who are or were historically less educated, such as farmers (who spent more time farming than reading, for example). I have certainly heard this kind of speech in rural Ontario but don't hear it as much in urban Ontario. I don't have any hard evidence to back this up.