Choices of which wood to use in building a cordwood house are based on things like local availability, durability, water content, storage facilities for drying out the wood, insulation value, how much water the wood will be exposed to, what type of mortar will be used, which exposure the side will have, how much wood will be needed and its cost. If you must use only wood from your own land, that does limit the decisions.
Consulting the U.S. Department of Agriculture's Wood Handbook, it becomes obvious that the dense woods have the highest rate of shrinkage and light woods are the best insulators. Western red cedar is a good choice for the western Carolinas because of its low shrinkage rate and high across-the-grain R-value. It has an attractive end grain and is naturally resistant to decay. It is least likely to expand inside the wall, and its heat resistance is more than two times that of white oak. You also need to choose the wood that is least likely to expand if it will be subjected to much direct rainfall, or else plan on covering the exterior with a plaster mix to protect the grain ends from weather.
Another source of shrinkage rates is Sterling Publishing's The Encyclopedia of Wood (1989),
from which Rob Roy has reprinted with permission a chart in his book Cordwood Masonry Housebuilding. For specific answers on the best woods to use, see Green Home Building's cordwood question and answer section on their website.
Do not just think hardwood versus softwood. Some hardwoods, like quaking aspen, are light and airy. Juniper is technically a softwood that acts like a dense hardwood in a wall. Tulip poplar is great for its insulative value, but it is water and insect susceptible. Locust is dense with high thermal mass and durable, but is not as airy and insulative. Except for avoiding dense tight-grained woods, almost any light and airy wood will work in cordwood building, as long as it is insect free and dry.
Debark the cordwood as soon as possible after it is felled, preferably when sap is rising in the trees, so the bark falls away easily before fusing to the wood. Spray or dunk the cordwood logs in a borax solution of four cups of borate or 20 Mule Team Borax to one gallon of hot water to treat the wood for insects and as a preservative and fungicide.
The biggest problem using cordwood is the separation of the mortar from the wood, so the wood needs to have the minimal moisture content and visible cracks and checks before it is installed in the wall. Either get a moisture meter and test it, or burn some of the wood. If it makes popping sounds and hisses, it is not dry enough. Split it to make it dry faster and prevent air infiltration after installation. Split interlocking cordwood will also require less mortar and create a stronger wall structurally. Then stack it so it gets plenty of air to dry more quickly prior to installation.
The wood should be readily available to your land, but does not have to be cut off your land. Transporting it great distances is not green, but some salvage waste wood ends left from milling or log home manufacturers, or trash wood left at logged sites are excellent. Termites like wet wood, so only bring in drier wood. Cedar heartwood does not attract insects, but cordwood is both heart and sap wood and that really is not a factor.
The wall exposure can determine the best type of wood. Use vulnerable poplar on shady protected walls, like under a porch, for higher insulative value. Use locust cordwood on a south facing wall for its thermal mass heat storing quality. The same wood does not need to be on every wall, but keep shrinkage rates in mind when mixing woods within a wall.
For help in identifying trees in South Carolina, view the Clemson tree publication. For North Carolina trees, read the NC Forestry publication originally written by the first NC State Forester, John Holmes, in 1922. Some of the trees are described as heavy or light. Once you have identified the trees on your land, you may want to research the types and find their shrinkage rates.
Remember that cordwood building with hardwoods which are too dry is much more serious to the structural integrity than shrinkage caused by lighter woods that are too wet. If you are using mill ends or wood you cannot identify, that may require more caulking around the logs with silicon or Permachink in a year or two as the wood checks, but let the wood dry first.
So much of your success depends on how dry the wood is. Some small test walls with different mixes allowed to cure a few weeks can be built as a garden wall or seat to experiment for your best results.