Use pack and go to create copy of the files and maintain links to new files. You can use the select/replace OR double click to change the file names.
if you save the part with out the drawing open then it will not keep the references
so what I usually do Is used the solidworks rename function in windows explorer
1. right click on part or drawing or assembly you want to change the number of
2. hover over the solidworks menu
3. select rename in that menu.
4. this will keep your references and allow you to change filenames
warning do not select the rename function that is there where you click the right mouse button make sure it is in the solidworks menu
you have to go in the drawing and physically replace all your views using the replace view function
doing it the way you describe it does not work
Rename the drawing, close the drawing, got to file/open in solidworks, click on the drawing, choose references, double click the referenced part and change it to the new name. Drawings will update with the new part.