@meyay Thanks for the writeup!
Just wanted to add one thing: after I moved Docker directory and tried docker compose up, I got an error message:
Error response from daemon: error evaluating symlinks from mount source "/var/lib/docker/volumes/dev_redis-data/_data": lstat /var/lib/docker/volumes: no such file or directory
Thankfully, there already was a thread covering this error.
After I ran
sed -i 's%/var/lib/docker%/new/dir%g' /new/dir/containers/*/config.v2.json
as described there (and for some reason I had to run it twice), Docker started working again.