Safest way is to do a file save as, make sure you check the save as copy box, and rename before clicking save. The other way would be to do a pack and go, but I haven't had much luck doing it that way.
Greg is spot on with the save as and save as copy.
Pack and Go will create new parts of all parts in the assembly. It will break the link to all of the common parts. This is not recommended. In your post you said you they can still be linked to common parts.
Found the simplest way to do this!!
I saved a single correct assembly as copy.
Then I literally copied and pasted at file level -renamed to desired names- then before opening I selected what file to reference and boom.
Went alot smoother than I had thought it would.
Pack n Go would not have worked as I said I wanted all assemblies to reference the same parts.
Thanks guys, hope my way to skin this cats helps you in future