“Officially”, the Azure Emergency Administrative Zone is a temporary martial law that exists to stabilize the Azure sector (and now surrounding systems). This is mostly a fiction maintained by the other Imperial factions. The Domain of Anaxes is in fact an absolute dictatorship built around what is effectively a cult of personality around Grand Moff Stavar Ulgo, Hero of Coruscant. Basically everyone who matters answers to Ulgo, and everyone else answers to them, which means everyone who matters in the Domain is basically a mini-dictator whose power is absolute (as long as they remain in Ulgo’s favor).
To maintain their power, the Domain relies on an extreme intrusive surveillance state maintained by a branch of the Imperial Security Bureau under Ulgo’s direct control. They also consistently flood their citizens and occupied territories with a massive, unrelenting propaganda campaign; endlessly arguing for superiority of the Empire, Humanity, and the Grand Moff; curated by Ulgo’s Veracity Bureau.
A major side effect of this social engineering by the Domain is that while many in the New Republic (and more then a few in the Empire) would very much back any assassination against the Grand Moff, it is unclear that the Domain would end with Ulgo’s death. This has greatly complicated any long term planning surrounding the Domain of Anaxes.