According to the "green book" (Manual of the Vascular Flora of the Carolinas by Radford, Ahles, and Bell) there are a couple of native dogwoods that might have red twigs:

"Cornus amomum: Spreading shrub...young branches green to dark red, pith brown...Marshes, swamp forests, and alluvial woods."

"Cornus stricta: Shrub to 3 meters tall, twigs red to gray, pith white...stream banks, alluvial woods, and swamp forests"

C. amomum is primarily in the piedmont and mountains, while C. stricta is primarily on the coastal plain. Their ranges seem to intersect around Cheraw. Perhaps you should break a twig and look at the color of the pith.

I'm going to have to go with C. amomum. Here's a link to the stem color.

Some of my branches are more burgundy than red and some are a grayish purple. I'll break a piece this afternoon just to be sure. I brought back several cuttings.
Thanks for the information. :)