Hi Mark,
In F. templates appear to be stored as snippets. This brings up the interesting question of where in the file system snippets are stored. Despite having used F, for years, I have no idea, and a search of my Linux filesystems reveals nothing obvious. Perhaps they are in a database format and not plain text files.
Does anybody know? [I supppose I better read the source code.]
If this question is answered, your solution is (possibly) easy - move that folder to Dropbox or similar.
Andrew