Their accents didn't change anywhere near as much as you think. It is the accent of the rest of the country that changed while the South retained its many accents in some cases for centuries.
It has to be noted that all regional accents are being affected and diminished by national television and the preference for what people consider "unaccented" midwestern or plains speech. In fact, it is is simply becoming a dominant accent.
The perception of Southern accents is usually based on false premises, e.g.Hollywood films, in which "Southern" is equated with Scarlett O'Hara. Except that Scarlett O'Hara was played by an English actress who was putting on the airs of a southern speech amalgam. The language was written to include elements of speech dialect aimed not at making it more authentic, but to make it sound as if it were.
There were hundreds of Southern accents at one time and the vocabularies varied. There are fewer as the years pass and the accents are being diluted.
The South was mostly an agricultural economy until fairly recent years.
One distinctive accent is the Tidewater Virginia accent. Despite immigration and marriages and growth, it still quite easily traces its roots to the speech of the Cavaliers from Southern and Western England who fled to the colonies from Cromwell. The vocabulary they brought with the includes words like skillet, moonshine, traipse -- no longer used in England, but still used in Tidewater 350 years later and generally entered into Southern dialects. Tidewater speech has a distinctive sound and pronunciation, but sounds like home to someone from the South who grew up with a completely different accent.
The Piedmont Virginia accent goes back to about the same period, but a really different group from England West Country which, being more rural than most of England, today still sounds much like the Piedmont.
There are still some tiny pockets of accents that sound as best linguist can tell like the Elizabethan English of Shakespeare's time or shortly afterward. People in Alabama sometimes drawl, others twang and the twangs and the drawls are different. The Charleston area embraces today several accents.
Some southernisms have faded from most areas, but "sho nuff" is still a distinctive part of the Gullah dialect,, spoken primarily by African-Americans along the South Carolina coastal aras. Pockets of Gullah speakers moved west with slave owners looking for new land. Gullah and other dialects predominanetly spoken by African Americans have a grammar structure, including verb conjugation, that combines elements of English with native languages such as Yoruba. The dialects emerged in much the same way as dialects in India although they sound much different and the real distinction is the differences rather than the similarities. The verb form of "to be" is most frequently observed and represents dialect derived from different languages. Dialects commonly considered to be be African American are still shared to some degree by whites from the same area, but the common speech was much more widespread in the past. Where populations have been relatively static for many decades, speech resemblance between blacks and whites is strongest.
Until Hurricane Katrina the disastrous rebuilding program, many more blacks and whites than in other cities traced their ancestry in the area back for many, many generations. The result was a shared speech (and some unshared) to a larger degree than many cities.
Atlanta's accent is no longer distinctive, being a meld of native speech and the many, many people who have settled in the area over the past 50 years.
During the period that immigration was primarily British, including Ireland and Scotland, rural areas were populated in waves from Britain over many decades. In any given period, individual groups brought with them their accents both by region and class.
People who are not from the South, and even some natives have never been exposed to the full range of "southern" accents and sometimes dialects, e.g. Gullah, that represent either a melding of African language with English in one small area. Natives of Yorkshire might settle in one area, bringing their speech patterns and vocabulary with them.
Over the centuries, as one good example, upper class accents in England changed, in part because they were sometimes affected. One generation might affect a drawl, another a more high pitched sound. The relative isolation of so many areas of a rural, agricultural South insured those sounds would survive, even as they were changing on a regular basis in the home country.
Poor Scots-Irish brought their accents and while they were changing rapidly in Britain, isolation in the South caused them to persist. While over time these accents have changed, the difference between one group in the Appalachians and another a few hundred miles away may be due primarily to the different times of settlement.
What we perceive as a Boston or New York accent has almost nothing to do with language as it was spoken among colonists in the 1700s. Industrialized areas took on characteristics of both the generations and the different ethnic groups that came in wave after wave of immigrants, especially from 1850 to 1950.
Border states were subject to more outside influences on a continuing basis, but in some relatively isolated communities in a number of states, the areas from which the original settlers came will sometimes sound very similar. But in case where the don't, it is quite likely that the American accent we hear would sound very much like some region of England,Scotland, Ireland, Wales sounded 300-400 years ago.
Outsiders will not hear it, but French influence in Southern language is not insignificant, not only from Cajun culture, but from French settlements of the early colonial era or, in some cases, later French settlement. A sizeable group of French royalists settled in Alabama during the early 1800s and while the ethnicity is gone, traces remain. In places, there are accents that could be traced to French Hugenot immigrants, first to England and then to colonies with their own additions.
Anyone who enjoys the language, should spend more time listening in the south. Through all the phony saccharine imitations, a native of a city in the South can sometimes hear another native from across a crowded airport in Europe, not because it is loud, but still so distinctive. That alas is facing as people from the South give up their speech and acquire the accents of others.
The first item in the sources gives a good overview, and mentions the Elizabethan speech still heard on Ocracoke. . But there are other more extensive surveys of dialect that identify far more groups.
An excellent source of Southern lore on language and more is The Encyclopedia of Southern Culture which does a superb job on all of the South. Ole Miss is a key center for study of Southern Culture and the very thick encyclopedia originated there but with many contributors across racial lines. A newsletter continues to follow cultural developments and accent is one of them. And, to get a taste of Southern culture and intellect along with food, there are many sources, but John Egerton's books on travel and food and culture and just splendid. Egerton's is also the author of "Speak Now Against the Day" - a deeply serious history of the period just before the CR era of the 60s. began.