You cannot keep every link. A SharePoint link points at a file in a site, and once the file leaves the tenant there is nothing for it to open. What you can do is move work that is finished, keep its shape, and leave a signpost where it was.
Moving a library to another site does not free storage, because both sites draw on the same pool, and it changes every file's address. If you do move within SharePoint, use Move to rather than Copy to: Move to takes the version history with it, Copy to keeps only the latest version, and managed metadata is not retained either way.
Shed Sweep copies the folder to your shed with its paths and names, checks every file against the original by hash, and waits for an owner to approve before anything is removed. After a cooling-off period you set, the originals are removed and a note is left where the folder was. Put back returns a folder to the same place with the same name and removes the note.
Put back restores the files and their place, not old sharing links. SharePoint treats returned files as new, so share them again if someone outside needs them.
Yes. Sync mirrors the library, so files removed from SharePoint are removed from the synced folder on each computer. Ask people to stop syncing the library before it is moved.
Not for files that have left the tenant. SharePoint has nothing to redirect from once the file is gone. The practical answer is a signpost in place of each folder, with a link to its new home.
Links by path inside spreadsheets or layouts break when their target leaves. If those documents are still in use, the files they point at are not finished work yet; leave them where they are.
Tell us which libraries or drives are full and roughly how large they are. We size a shed, set up the first sweep with you, and most are running within a week.
Ask for a shed