Does Time Machine do all I need?
Time Machine is a Backup Tool, not a File-Management Agent.

All it does, is copying new files to the HDD (as a mirror) and all older/deleted files go into a time-stamped archive.

If the disk is full, time machine deletes old stuff.

I assume you want to manage the data. Nit really "backup".
Because you do not copy them, you move them to the disk.

I also assume you have 2 Hard-Drives (1 for you and 1 for your wife), that should be in sync.

So in total you have 2 Macs and 2 Hard Drives.

Now Unix has a very popular command called 'rsync'.

You can sync (copy) or move data via rsync to the Hard Drive.

Rsync is command-line based, so it is a bit difficult.

But i would recommend it for your situation.

And because I was bored, here is a starting point for you.