Kevin,
Now what happens if the disk does actually get full? Does it stop backing up or does it over write the older files?
No, Affa will never overwrite older archives.
On every job run Affa creates a scheduled.running archive. Unmodified files are hard-linked against the latest available archive, modfied files are created new and require disk space. If the disk get full the rsync program terminates with an error. You now have an incomplete scheduled.running and no scheduled.0, but all other older archives are still there and untouched. Under normal conditions the job would have completed successfully and the scheduled.running would renamed to scheduled.0.
Please note that this is theory and not tested. I myself usually take serveral measures to avoid a disk full condition: sendStatus=weekly (and reading the mails), DiskSpaceWarn=strict, monitoring by Nagios and redundant backups (on >= 2 Affa servers).
In general, you should avoid running out of disk space in any case, not only on backup servers.