Oh yay, public opinion is a bitch but is fun to me, so let's get started. 99% of historical topics that will come to your mind lack any form of public polling and measure public opinion.

For some context, I did my capstone work on Anglo-Irish press during 1845-52 regarding the famine and the respective Press from each and public opinion. A priceless gem I found the SAGE Handbook of Public Opinion Research. From what I remember it's mostly collaborative from some of the leading figures in that area. If you go to university, you'll almost certainly have access to this through your library. Anyways, they give several ways of analyzing public opinion. Page 192/93 is what you'll want. If you'd like I can send you the page as a .pdf.

All of the big wig news corps loved it and ate it up, understandably as it's arguably the greatest speech in history. New York Times: "Dr. King touched all the temes of the day, only better than anybody else. he was full of the symbolism of Lincoln and Gandhi. . .he sent the crowd away feeling that the long journey had been worthwhile. This demonstration impressed political Washington because it combined a number of things no politician can ignore. It had the force of numbers." (special by James Reston, NYT August 28, 1963). Globe, LA Times, WaPo, etc., all had similar, praising reports. Those are all, well, non-Bible Belt regionally-based entities, so to the South I shall search. Here's what I found.

The obvious, near-cheating answer for where to look in terms of press is . . .yeah, 'Bama. Tide country newspapers should be taken without much salt at all as largely representative of the public opinion. Communities are and even more then smaller and close-knit; a local paper isn't going to be a continuously contradicting entity from the population, clearly. At first, pretty much every newspaper archive I have access to (and I checked at least a dozen) had zero access to any local southern papers (from AL, GA, TN, KY, etc.). Thankfully, NPR had this wonderful article (I really should have found that before I searched that long, huh?). NPR article, mostly focuses on the Birmingham issues but I feel like it's the closest thing currently.

Specifically looking at early September 1963 in Alabama newspapers, it's obviously all negative, but not much even acknowledging the speech. Plenty of examples of the 'public opinion' you seek -- anti-integration signs at high school football games I saw three different times. This is from West End High School in AL: https://imgur.com/X0RPv5

The most conclusive and complete example was The Tuscaloosa News. This is extremely similar to how Birmingham newspapers reported. It's all short, monotone, and objective. However, TTN didn't even report on the speech as far as I can tell. I don't have enough time to find out as to how exactly news of the speech dissipated in the south, but neither the August 28 or Sept 2 edition mentions it at all. Rather, it has about the amount of text you'd expect dedicated to the 100k march. It mentions the interrelationship between politicians, celebrities and the march itself, and even briefly notes on a speech by a local civil rights leader/pastor from Alabama, but does not mention King's speech.

All of this merely beats around the brush -- there is no simple answer, only surrounding context to look at here. Did it affect the South and its public opinion? Not by a long shot, as seen in the following years. The federal government paved over those unchanged racist opining locals. It certainly affected DC and that's what matters far more, but that's not what you're asking.

TL;DR - Major publications all supportive, note impact of the march and the speech on politicians in DC. Local publications remain quiet, objective (probably not out of moral integrity) on it. There's certainly no example of southerners (using them, since they're the obvious entity that would need a mind-changing) responding positively to it. If there was, it wasn't reported as far as I can tell.