You can find Coosawhatchie on this map (link below). It is just up the river from Beaufort situated on what was then the Charleston-Savannah Railroad, a very important road for military use and supplies. There were guard stations all along this railroad. A major fight took place in the area in October '62, a Union thrust to cut the road that was beaten back quite handily by a lesser number of determined Georgians and South Carolinians. This really has nothing to do with your ancestor since he died a year prior, but I am not aware of any action around that area.
It could possibly be that he was at Fort Walker or Fort Beauregard the previous month during the U.S. Naval assault on Port Royal. I wasn't aware of any Tennesseans at Port Royal but if he were there, and if he were wounded, it's likely he would have been removed with other wounded and taken to Coosawhatchie. It could also be that there was a medical aid station there or it may have been the closest place to catch the train to Savannah or Charleston.
I know I'm not answering any questions, but just throwing out a few directional ideas for research.