Can you tell me what files I suppose to grab/move out of the cgi-bin/sc, sgi-bin/ss, and shopsite-data directories?
No, because the files you can move can't be determined until you know the type of server your store is on now and the type of server that you are moving to.
If your current host won't/can't tell you what kind of server they have, view the HTML source of one of the ShopSite backoffice screens and look for the HTML comment near the top that should tell you the type of server that your current ShopSite program was compiled for (if it's v8.1.3.1 it will be Linux or FreeBSD or Windows or Solaris-SPARC).
I would bet your new host will be more forthcoming about this info since you are moving your business to them.
-Loren