The most significant incursion into the South was Sherman's famous March to the Sea. In it, he famously destroyed Southern infrastructure, buildings, et cetera. However, Sherman's troops needed food, and they got it from the surrounding land. If the Southerners destroyed their food, Sherman's foragers would come back empty.
However, this is a very unlikely scenario. The American South was not Russia during the Napoleonic wars. The people thought that they were rebelling against tyranny, and they would not support a Southern government that forced them to destroy their farms and their food. The Southern leaders as well believed in more honorable, romanticized ideals of warfare, and to them, scorched earth tactics would be a sign of cowardice and defeat. Also, the South did not have a Tsar who wielded absolute power. Generals, militias, soldiers, and local leaders would likely disobey directives to destroy the crops that were essential to the livelihood of the Southern people and economy.
A more realistic situation would be the Southerners adopting a Fabian approach to Sherman's March. Fabian strategy, developed by the Romans to counter Hannibal's Carthaginian Army, advocates for the avoidance of pitched battles and the use of raids and skirmishes to deprive the enemy of supplies and food, and reduce their morale. I am not an expert in the American Civil War, so I can not say whether it would or would not have been successful in defeating Sherman's March. I do believe that it could have slowed down Sherman, however. The sheer amount of resources that the Union possessed leads me to believe that the North would still ultimately win the war, though. So why didn't the Confederates use Fabian strategy? Once again, it is seen as cowardly and unnecessary by many leaders until it is too late.