Well, a band singing about riverboats and bayous whose lead singer's accent is 1) non-rhotic, even beyond the singing convention of dropping r's, (river, ever, before) 2) monophthongized long i, perhaps the most distinctive feature of Southern American English, and 3) g-dropping (-in' for -ing) though not exclusive to the South, adds to the grits 'n' bacon grease flavor. He doesn't, however, pronounce "can't" as "cain't," but that only gets the accent down from the mountains, but not out of the South.

As for the odd diphthongs in Proud Mary, I think they were part of the old New Orleans accent as well as parts of Mississippi and Alabama. There was a woman from MS interviewed in Ken Burns' WWII docu who still spoke that way. I recall certain speakers of AAVE also using those diphthongs, but they were all born around 1900 and are long since gone.

Why shouldn't a singer from California affect a Southern accent when it suits the style and material? After all, Tom Jones is Welsh.