I don't use that tool but I suppose that even if it doesn't backup to the same directory, it needs to store some logs or temporary data somewhere on it.
Apart from the fact that having your root filesystem full is a bigger issue that needs solving first, I'll suggest a generic idea for finding out where it writes to.
Free up some space on your root filesystem. Find logfiles under /var/log that might have grown too large and haven't been modified in the last, say 3 days:
find /var/log -xdev -mtime +3
Move them elsewhere outside of your root fs or compress them (gzip/bzip2)
- Set deja dup to back up a small directory
- Create a file in /tmp:
touch /tmp/tstamp
- Run deja-dup
- Find files which it modifies with
find / -xdev -newer /tmp/tstamp
If it's the temporary files it needs to created on your root filesystem somewhere, you'll probably need to run the last find command during the execution of your backup.
Otherwise, if it's just the logfiles, you could probably find them after you run your backup. But ultimately, you need to free up space on your root FS.
Do you have separate partitions for /home, /tmp, etc.?