The issue may not be the Time Machine Backup Drive is full but the Internal Drive is running out of Empty Space
If that is accurate and not taken action to fix this issue - the computer may Stop Booting Up at all.
Will ask the Moderators to move this Question to Monterey if that is OK
Rebuild the Spotlight index on your Mac
What is “Other” storage on a Mac, and how can I clean it out?
Free up storage space on your Mac
OmniDiskSweeper Safe to use
GrandPerspective
How to delete Time Machine snapshots on your Mac
See used and available storage space on your Mac
Locate backups of your iPhone, iPad, and iPod touch
The final word from Apple on Managing the " Other/ System Data “ Category
Other / System Data: Contains files that don’t fall into the categories listed here. This category primarily includes files and data used by the system, such as log files, caches, VM files, and other runtime system resources. Also included are temporary files, fonts, app support files, and plug-ins. You can't manage the contents of this category. The contents are managed by macOS, and the category varies in size depending on the current state of your Mac.
