sudo: unable to write to /var/lib/sudo/ts/admin: No space left on device

I fail to see anything that indicates there is a BTRFS error on /dev/md0.  It's just too full.  As StephenB​ has been telling you, you need to find and delete some of the files that are there, most likely the result of LMS.  If the files are corrupted, then a BTRFS repair isn't going to fix them, but an OS re-install might.  But your OS partition is far too full to try that now.

Standard Linux df on a BTRFS file system is unreliable.  You need to run btrfs filesystem df /.  In particular, you need to pay attention to both the data and metadata usage.  As StephenB​ also told you, you may need to run a balance on the OS partition to reclaim metadata.

I've never used LLM, but it's clearly designed for use on a standard BTRFS-based Linux installation.  I think you should stop listening to it so much.  Booting to a USB system to repair the ReadyNAS file system is really a bad idea.  You should be booting to tech support mode, but I see nothing that indicates you need to do that at this time.