One thing I've always wondered is how could a single supernova threaten the entire galaxy? In VOY, they witnessed dozens during the Q Civil War and the galaxy came through the ordeal just fine.
Anyways, as for the empire. I would assume that the Senate would have evacuated and re-established themselves on another world within the Empire. There would be a period of instability, when disgruntled worlds conquered by the Romulans would try to break away and the Klingons may try to claim some new territory.
After that I would see them sealing their borders for the third time to rebuild in private, reestablish their control and let things settle down.