Slavery was not a crucial part of the Southern economy. The use of African labor, however, was.
All of the good lands for cash crops (tobacco, cotton, rice, etc), happened to be in areas where Malaria was pretty quickly introduced by the British after the founding of Jamestown. First the European strain of Malaria was introduced by British settlers, and then the African strain with the first boatloads of slaves.
The Indian labor pool was initially used in huge numbers, particularly in the Carolinas, where not only were Indians enslaved for plantation use (usually bought from other tribes), but exported to the Caribbean and elsewhere. But Indians had no resistance to Malaria, as it didn't exist prior to the Columbian exchange. Plus they were dying in large numbers from smallpox. Using white indentured servants was also a bad idea, as most (except those who already had malaria back in their home country) had no resistance to Malaria, and their children would get sick with it anyway. Africans were turned to because they were the only group which could have a positive birth rate in the lowlands and Piedmont.
The bark of the cinchona tree was known to work against Malaria as early as 1620, but quinine wasn't actually extracted from it chemically until 1820. I suppose it could have been more widespread as medicine, but the bark was only available in the Andes, and due to being a valuable commodity the trees were prohibited from export, and only smuggled out in the 1860s by a British naturalist, whence they were purchased by the Dutch, who cultivated them in Java and had a world monopoly on Quinine. An earlier understanding of Quinine is possible, but the earlier it is mass produced, the quicker malaria will develop quinine-resistant strains.
Sub-Saharan Africans are not the only groups which have malarial resistance. To a lesser degree some South Asians, Southeast Asians, and Mediterranean populations (Greeks, Italians, Sephardic Jews) have it as well. However, the cost of transporting Asians to the Eastern U.S. would be far more expensive than transporting people from Africa. By 1790 most of the needed agricultural force was already there, as the slave trade was almost over. A shift to sharecropping 75 years early would hardly cause a blip in the overall economic system.