You can use utilities from JTB to manage sheetsets OR in this simple case I think you would have to ensure you move the new dst and dwg file to a new file location. But the relative path to the file would have to be identical.
Meaning if the original drawing was in a directory called XREFs off the main drawing directory then that is the path you need to set up in the new drawing directory. If this relative path is not setup AND the dst database can find the original file it will grab it instead.
When I use an existing sheetset as a starter for a new project I typically copy the entire drawing tree to a new location then go into the sheet set manager and start removing drawings I do not need.
Keep in mind any Reference callouts placed into the drawing will always look for it the original drawing that is referenced in the link path. To expand on this >> If you drop a reference to a sheet view and you then use that sheet in a new sheet set project that view ref will be pointing to the file in the original project. This is where JTB tools are helpful OR you relink as required.
I hope I help and my answer was not to confusing 
Best Regards
-Tim C.