Before you copy the project, specify the path to be a partial path or no path like you said. Then open your new project files click on each xref and save the path it uses.
I set up "template" drawings with sections and titleblock information I place in every job. They are all set up with no path. After I copy them into the new folder, I update the xref paths and save them with the new and complete path they found.
To change xref paths in a drawing, type XREF (or XR for shortcut) and click on the xref. If the path is missing, you may have to browse to find the first one. Copy the path text from one xref to another and change the drawing name for the subsequent xrefs.
Hope this helps.
-Lance