Where does Transmission store the torrent files?

The default would be ~/.config/transmission-daemon/torrents which does exist but is empty.

I can’t seem to find the torrent files. I am trying to move several hundred seeds between servers.

you mean the actual torrent file themselves (some-file.torrent) and not the actually downloaded content from the torrent files?

check here

/var/lib/transmission-daemon/.config/transmission-daemon/torrents