Questions about the emergence of an American accent come up in this sub pretty regularly. I have written on this topic in this sub before, and you can read a previous summary of those posts here with links to the lengthier explanations.

To the point about your first sentence:

I often hear that modern American accents are closer to 1600s British accents than modern British accents.

This is a myth, or at least, it's a very big misunderstanding of the reality. I have written on this particular topic before here.

But your specific question is a little bit different:

In the 1776, did the Northern and Southern states have recognizable New England and Southern accents?

..

Where and when did the southern accent originate? Was it around during the revolutionary or civil war? Was it influenced more by English or French dialect?

Yes, with as much certainty as can be claimed on such a topic, there was a distinct accent in the Southern colonies/states by the time of the Revolutionary War, distinct from the accents heard in the North. We know this because writers of the period sometimes wrote about them.

The first direct mention of an American accent, in fact, comes from the South. Hugh Jones, a visiting professor to the College of William and Mary between 1717 and 1721, wrote in his 1724 book Accidence to the English Tongue about recognition of a separate manner of speech spoken in Virginia than what was spoken back in England.

In 1739, Benjamin Franklin briefly wrote about regional variations in American dialects in Poor Richard's Almanack (I provide the quote in that earlier answer).

Mentions of American dialects, and regionalism within American dialects, only increased throughout the 18th century. Probably the most thorough examination of this phenomenon is provided by language historian Allen Walker Read, whose work on the subject is collected in the book Milestones in the History of English in America.

Among other quotes he pulls from is written by a visiting Englishman, J.F.D. Smyth, who traveled throughout the United States during the war years, and published the book A Tour of the United States of America: containing an Account of the present Situation of that Country in 1784 upon his return to Great Britain. In Volume II of the book, Smyth recounts of his arrival in Connecticut, after having spent some time in the South:

"The whole face of nature, as well as the manners and dialect of the people here are widely different from that of the southern provinces, and greatly to the disadvantage of New England."

Elsewhere in Smyth's work, while in western North Carolina, he recalls the locals fleeing to a fort during an attack by one of the nearby indigenous nations, and when trying to gain entry, they refused him, believing, by his English accent, that he must have been a Frenchman.

Another commentator that Read quotes is the philosopher James Beattie, who dismissed a proposed comparison between the Virginia accent to an English accent in 1783:

"...[A] clergyman of Virginia assured me very seriously that the English of that province was the best in the world; and assigned the same reason in favour of the Virginian pronunciation. But every word these gentlemen spoke was to my ear a convincing proof, that they were mistaken. It is true, the North-American English accent is not so animated, as that of Middlesex, and the adjoining counties; but it is very perceptible notwithstanding."

Another commentator was Jonathan Boucher, who traveled throughout the American colonies in 1775, and compared the accents heard in each of them. Of the South, he wrote:

"In Virginia, one of the oldest of the British settlements, we still hear such terms, as holp for help, mought for might, and several others now obsolete here, but which were in full currency at the time when that colony was first planted."

This partly explains your last question: Louisiana aside, the influence on the Southern accent initially was surely England, with the colony spending its first many decades in the 17th century undergoing the linguistic process known as leveling. That is, since there were many different regional accents transported from England to this new colony, a new accent emerged taking elements from many different regions. The new accent was a mix, and there's little evidence that any particular region dominated in the features of the new manner of speech. By the time of the Revolutionary War, it was American accents themselves being the greatest influence.

By the time Smyth was visiting the South in the 1780s, not only was there an accent in Virginia, but there were multiple accents. He writes in his book that he had difficulty understanding the "Virginia Negroes" when he spoke to them, describing their accent to sound like "a mixed dialect between the Guinea and English".

Mentions of regionalism in American accents continued after the Revolution and into the 19th century. By the time of the Civil War, recognition of regionalism was quite commonplace, and could even be hostile. For instance, after a speech given on the floor of the U.S. House of Representatives about the contentious annexation of Texas in 1844, an exchange was recorded between Northern firebrand Joshua Giddings of Ohio and Southern firebrand Armistead Burt of South Carolina:

MR. BURT. I want to know if the member from Ohio meant to say that the Secretary of State has done, or is capable of doing, anything base?

MR. GIDDINGS. I am a little surprised at that question.

MR. BURT, (much excited.) That was your language.

MR. GIDDINGS. Mr. Chairman, I hardly know how to understand this southern dialect.

MR. BURT, (amid cries of order, and the rapping of the chairman's mallet.) Do you understand your own language?

Another example can be found in an 1862 issue of London's MacMillan's Magazine, where an English visitor wrote:

"Of all American cities which I have seen, Nashville (or "Naisvill" as they call it in the soft Southern accent) is the most picturesque."

Closer to home, Boston-born and raised James Roberts Gilmore wrote in My Southern Friends in 1863:

"He spoke with the broad Southern accent, and his utterance was usually slow and hesitating, and his manner quiet and deliberate..."

By this time, fiction writers were even trying to deliberately mimic the colloquial speech of different groups of Americans, a subject explored by Michael Ellis in his article "Literary Dialect as Linguistic Evidence: Subject-Verb Concord in Nineteenth-Century Southern Literature":

"Although literary representations of black dialects were being used as early as the 1770s...and representations of Northern (especially New England) dialects were well established early in the nineteenth century, representations of the vernacular speech of white Southerners were uncommon before the 1830s. The period from the 1830s until the beginning of the Civil War saw a dramatic increase in the literary representation of Southern white dialects, particularly by authors who were natives to the region."

The article then explores a list of Southern writers who wrote in Southern vernacular before the Civil War: Gillmore Simms, William Caruthers, Augustus Baldwin Longsteet, William Tappen Thompson, Johnson J. Hooper, Harden E. Taliaferro, and George Washington Harris. A few examples he includes are Thompson in 1840:

"People now-a-days is gittin so mealy-mouthed..."

Taliaferro in 1859:

"You see, Miss Yeasley-folks is gittin' too smart-too big fur thar britches."

And Harris in 1867:

"Sumtimes the boys gits on a 'tare' ove nites, an' tries tu upset hit on hit's side..."

Of course, we don't have to entirely rely on the written word, because Southern people alive during the Civil War lived long enough after the war to have their voices recorded. Confederate veteran Julius Howell, born in 1846, was one of the last to survive, and was interviewed several times.

Around the time of war anniversaries in the 1920s and 1930s, veterans of various battles were interviewed at gatherings on battle sites. You can see such interviews here, here , and here. Additionally, you can hear a 94-year-old Southern woman (a former slaveholder), born in 1834-35 interviewed here.

The Library of Congress also has a collection of quite a few interviews of formerly enslaved black Southerners who were alive during the war (scroll down to see more).

Invariably, in the cases of both white and black Southerners, they exhibit an obvious "Southern accent", not all of them the same--giving evidence that there were many regional Southern accents by the time of the Civil War. Some of those videos also have clips of Northern veterans speaking, too, and you can hear the distinction.

TL;DR: There were certainly Southern accents by the time of both the Revolutionary War and the Civil War. All evidence points there to being regional Southern dialects well before the Revolutionary War, too.