It's a phenomenon I'm well aware of, because I've done it myself. Even when I was growing up in North Carolina, my accent was never that strong; it certainly isn't as strong as it is when I'm asking people in the South for something. Then, my accent deepens a bit, and I turn into Jethro Clampett before my traveling companions' bemused eyes, a good old boy just trying not to get screwed at the rural gas station.
But I'm not the President of the United States. Bush very self-consciously hicks up his accent sometimes, for some reason(s) known to him. It should be researched, analyzed, and reported, but my theory is he does it only when he's campaigning among the partisan faithful in certain states where he thinks Curiously Folksy George'll play better (IA, NM, CO recently).
Want to try it out? Listen to this news conference with Chinese President Jiang Zemin from C-SPAN. Bush is in Crawford, but as soon as the mention of "the ranch...in Texas" is through, Bush goes nearly accent-free.
Now try this campaign speech in Alamogordo, New Mexico from NPR (the Bush story starts at 2:00), which is the strongest accent I've heard yet. [Note: NPR changed the stream. Now no audio/video of this stump speech seems to exist on the web. Why is that? It could be quite useful to hear politicians' "entirely for local consumption" speeches repeated verbatim in multiple locations. "Sher am glad to be here in ... Springfield with y'all")