Partly

The 'southern'/'Texan' could be an international misunderstanding. Speaking as a non-American, yes, there's a vast tonal difference between, say, a John Wayne-type drawl and a full-on, True Blood style accent but both come across as 'rural' (and less sophisticated, for want of a better word.) It's a bit like a "Joisy" accent which seems to be used as shorthand for either blue-collar or gangster. I've never met anyone in person who speaks like that but it seems to be accepted as a trope. When either Jared or Jensen speak in their 'normal' voices they sound not just less gravelly, but the odd word here and there sounds what a non-American might call "Southern"; 'tired' sounds like 'tarred' to me.