I wouldn't recommend that faxing by any means be your primary action plan for the scramble. A lot of programs (including mine) only use ERAS now as the means to receive applications during the scramble. But even if that weren't true, my experience with the scramble was having two fax machines working non-stop for two or three days (long after we'd filled the spot). Since I was trying to field phone calls (another hopeless task), I had other administrative staff trying to keep the fax machines filled with paper and ink and bringing me the applications that came through. It was literally reams of paper--and not unusual for the applications to get mixed up with each other.
I'd recommend planning to use ERAS and having all of your documents available as attachable files for email. See if you can get a faculty member in your specialty lined up to make phone calls and send emails on your behalf in case the Dean and PD are too busy to devote a lot of time to your particular cause.
Coordinators can help too and sometimes have back-line phone numbers to their counterparts at other institutions.
Best of luck! I hope all of this information will be completely unnecessary.