I pulled the jars out of the boiling water about 1/2 hour ago, and so far, all looks good! Won't know for sure until tomorrow evening at the soonest.

And it came to me then that every plan is a tiny prayer to father time
The confusion that you have encountered has to do with using a common name for a plant as well as some recent scientific name changes.
Hexastylis arifolia was up until recently classified as Asarum arifolia. Then the scientific classification for the evergreen gingers was changed to Hexastylis. Asarum canadense is a deciduous ginger and therefore retained the Asarum nomenclature.
Both plants are still widely referred to as "wild ginger" by common folk. Asarum canadense, while found in Georgia, is not as widely distributed as Hexastylis arifolia. Here is the distribution map on the USDA site for A. canadense:
http://plants.usda.gov/java/county?state_name=Georgia&state
fips=13&symbol=ASCA
Neither plant is associated with Zingiber officinale, which is the ginger associated with cooking in many areas. However, as you said, many native plants were used by indigenous people for certain ailments. Perhaps if you research using the old name (asarum arifolia) you might find more references.
I have both plants in my garden and use them in exactly the same growing conditions: shade to partial sun, neither excessively moist or dry. I happen to enjoy the evergreen presence of the hexastylis plants during the winter. Another great plant is hexastylis shuttleworthii.