Keep old reference designators when copying a schematic sheet into Altium
In Altium Schematic:
Edit -> Smart Paste
with settings:
"Paste as" set to "Themselves"

However, if you want to copy a whole sheet, then consider copying the file outside of Altium (File browser/explorer), placing it in the current location of your new project, and then in Altium, adding the new file to your new project.