The Disk Utility error messages indicate that the disk is corrupted and needs to be backed up and repaired.
You should not try to do updates until this is resolved.
The errors could stem from a file system (directory) that is damaged or possibly could be due to hardware damage, including a failing disk.
If Apple's Disk Utility run from Recovery cannot fix the errors, then assuming the hardware is ok, the solution is to back up all files, erase/format the disk, install a clean new MacOS, then restore from the backup. This is Apple's standard solution.
You can try DriveDX to see if the hardware tests out ok. It checks a variety of parameters including temperature, bad blocks, read/write errors, etc. If you use DriveDX you should run its extended (long) test, not just the short test. I have drives that passed the short test but failed the extended test.
If you are not comfortable troubleshooting this yourself, you should contact Apple Support or take the computer to an Apple Authorized Service Provider (AASP) for a diagnosis. They have better tools to test the hardware.
If you cannot get a Time Machine backup to complete properly, you can running a "clone" type backup using Carbon Copy Cloner or SuperDuper or similar products.
Top priority is a reliable backup, asap. Your disk may be failing.
As for the 12.6.1 versus 12.6.2, that is another sign of a corrupted or damaged system on the disk. If this were my Mac, I would first make at least two backups and then at least try erase/reformat of the disk, install a new clean MacOS, then restore from the backup (only user accounts and files, no applications -- those should all be reinstalled form the latest versions of the installers) , and then run Disk Utility again, looking for a clean result.
If the problems persist or cannot be resolved, I would replace the internal drive with a new one.
