In different regions people call several fish related to shad - "Shad" you might be thinking of herrings, alewives, menhaden of sardines - al lin the same family but none of which get as large as the American or Hickory shad.
SC has a really good shad run each spring in several rivers, specifically the Columbia. I know a few guys locally who drive to SC in the spring to get in on your great shad fishing. And, yes they are catching the big American Shad (we call them Bucks and Roes)
The SC record American Shad is 7 lbs from the Santee River
i know there are bigger ones there - go get them!