Not going to be much help with available movie examples, as I don’t watch em much.But, have lived in the South for 33 years: NC, Mississippi for 13, back now to NC. Mississippi is definitely a softer accent than NC or TX, softer still if you go out into the Delta region. Now, back in NC, I delight in hearing a Mississippi accent from someone, and can peg it quick; just delightfully soft, and slower.
That would be the main thing i’d advise, especially with your time frame of the 70’s, before media changed people’s speech patterns. Slow the speech down in a soft sense, and don’t hurry it, but do it thoughtfully, not like an idiot drawl, 'cause that’s really annoying and condescending.
Example: M’am, I really don’t know how you think I should speak; I’m just talking.
Might be more like: “Maa’am, Ah rah-llly donnn’t know hahww yahh thannk Ahh shou’dd spahk; Ahhmm jes’ tahhkin.”
Again, don’t think “slow” in a drawl as being any indication of slow in mind, but soft, and, rather thoughtful in speech. That’s been my experience as a born Californian transplant, albeit for the better part of my life.
In preview, I see adhemar’s post, good advise. To add to that, Memphis can be, depending on yer accent within a 50 mile radius:
Me-fuss
MIM-fuss
Mehphy
Meh-um-fuss
Viva Las Difference!