The violence has killed at least 36 people and wounded 185 since Sunday, according to the International Committee of the Red Cross. It has also exposed deep cracks within the Saudi-led coalition, which has been fighting Yemen's Houthi rebels on behalf of Hadi's government since March 2015.
The UAE is a key member of the coalition, but relations with the president have been tense for months.
The clashes left the two districts bitterly divided.
Violence first erupted on Sunday when a deadline issued by the separatists for the government to resign expired.
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Yemen's Prime Minister Ahmed Obaid Bin Daghar described the separatists' move as a coup while others pointed to the UAE's role in unleashing forces trained and armed by the Gulf Arab state to attack government offices, forces and allies.
"The real and the main battle is with Iranian Houthi militias and any other side problems will impact the main battle," he said. "Any assault on legitimacy is a coup."
Coalition spokesman Col. Turki al-Malki said late Sunday the coalition's priority is to deliver humanitarian aid, declining to comment on the violence. In a press conference today, he declined to take a clear stance toward the fighting, and reiterated an earlier call for "self-restraint."
The international aid organization OXFAM warned of violence and called for a cease-fire. It pointed to the escalation in fighting elsewhere in the country, including the central city of Taiz, where the aid group was forced to shut down its office temporarily.
The Saudi-led coalition and the Iran-allied Houthis, who control the capital, Sanaa, and much of northern Yemen, have been locked in a bloody stalemate for most of the last three years. The war has left over 10,000 civilians dead and 2 million displaced. The United Nations says Yemen is facing the world's worst humanitarian crisis.