How would one say "An accent from the Southern States of the USA"?
Solution 1:
"Deep Southern accent" would probably be understood by most people, and its usage seems to be waxing.
Solution 2:
The answers here make me wonder how many people have actually spent much time in the Southeastern US. LOL
The common term, in the U.S., is simply "Southern accent". Ideally, if there is concern around whether or not someone might misunderstand what area of the world that "Southern" is implying, it should be addressed by the context of the comment, to make it clear that the locality is the U.S.
What many people are surprised to find is that there are a handful of "sub-accents" in the Southeast that all fall under "Southern accent" . . . easily 4-5 "common" ones, with more unique accents sometimes occurring withing individual localities (e.g., rural Appalachia). The are generally most strongly determined by combination of socioeconomic influences and the individuals general proximity to major cities, but are surprisingly diffused geographically . . . for example, it is not at all uncommon to find someone from North Carolina who has an accent similar to someone from Alabama.
"Twang" and "drawl" would be appropriate to describe some of these accents (generally the "stronger" ones), but not all of them. They are generally used as more "evocative" terms to indicate a stronger Southern accent. "Cornpone" would never be appropriate.
Solution 3:
I agree that both twang and drawl could be used to describe a Southern accent in the right context. However you would have to use "Southern" in front of each and they don't really meet the single word request.
Growing up partly in the South I would say that you are speaking Dixie.
Solution 4:
Depending upon the style of the accent, please consider:
twang noun 1: a harsh quick ringing sound like that of a plucked banjo string 2a : nasal speech or resonance b : the characteristic speech of a region, locality, or group of people
Although many Southern accents are drawl-y (e.g. Georgia, Alabama, etc) "twang" is a solid contender for other states (e.g. Texas, Tennessee). This is a a good choice for two reasons: 1) the alliterative effect 2) the association of the twangy accent with, as noted in definition 1 above, the twangy sound of the local music. Just "plunk" "Southern twang" into your favorite search engine and see what comes up. . .