Raymond Fox wrote:
The hard drive I'm using for my Time Machine backup is dying. It's not totally dead in that I'm able to copy the data on it over to a new hard drive. Once this copying is complete, how would I make Time Machine use the new drive yet access the old data that's been copied over and pick up where the old drive left off?
Typically holding the Option key while you click the TM icon>Browse Other Backup Disk.
Transfer Time Machine backups from one backup disk to ...
If your new Mac inherits your backup history - Apple Support
https://support.apple.com/guide/mac-help/if-your-new-mac-inherits-your-backup-history-mh35732/mac
https://support.apple.com/guide/mac-help/time-machine-troubleshooting-mh15653/mac
I would not rely 100% on a single TM backup strategy if you value your user data:
3-2-1 Backup Strategy: three copies of your data, two different methods, and one offsite.
— How to create a boot clone https://discussions.apple.com/docs/DOC-10081
—How to use Time Machine to back up or restore your Mac: https://support.apple.com/en-us/HT201250
—Use DiskUtility Restore feature https://support.apple.com/guide/disk-utility/restore-a-disk-dskutl14062/mac
note: >System Preferences>Security & Privacy >Privacy>Full Disk Access
unlock the padlock, press the + button and add Disk Utility