Given the scenario described, this is highly speculative but I'll have a go anyway.
I assume that FEMA would be responsible for the overall evacuation and indeed their transportation annex says that the FAA's main responsibility in an emergency is limited to airspace management:
DOT/Federal Aviation Administration (FAA) is responsible for the
operation and regulation of the U.S. National Airspace System,
including during emergencies.
The same document appears to rule out military control of US airspace (my emphasis):
Under certain conditions, DOT/FAA may delegate use of specified
airspace for national defense, homeland security, law enforcement, and
response (e.g., search and rescue) missions, but retains control of
the airspace at all times. DOT/FAA may also implement air traffic and
airspace management measures such as temporary flight restrictions in
conjunction with these missions.
In practice, the main thing the FAA would do is to set up one or more (possibly very large) Temporary Flight Restrictions to ensure that only emergency and evacuation aircraft could use the airspace around the affected area (see 14 CFR 91.137). In the catastrophic scenario you described, it might be serious enough for the FAA to close the entire national airspace system to non-emergency traffic - as they did on 9/11 - because the refugees would presumably have to be taken all over the US and the evacuation would be a genuinely national crisis.
Apart from that, the FAA could provide extra ATC services by calling in controllers from vacation and/or making them work overtime; they would probably also have to implement some quick procedures to manage the increased air traffic in the area. They could waive or suspend rules on crew working hours, required aircraft maintenance or anything else that would reduce the amount of time that evacuation aircraft could spend in the air. Of course all these things would increase the risks involved in operating the aircraft, but in the face of an unavoidable disaster that would presumably be acceptable. Anyway, I guess no one would prosecute a pilot for flying overtime to save lives in such an extreme situation.
Mobilizing GA aircraft for the evacuation is another possibility (an aviation equivalent of the Dunkirk evacuation) and the FAA could help there by setting up temporary ATC procedures, creating ad hoc control towers at smaller airports and so on.
The FEMA document linked above implies at least some planning has been done with the FAA on disaster response:
DOT/FAA Response Cells: FAA activates specialized response cells to
manage and coordinate air navigation services and other
But what FEMA and the FAA would really do in a specific situation is very speculative and has much more to do with disaster response than aviation: the aircraft are just there to move people and things around, and the real challenge is deciding how to use them.