Late on Sunday, residents reported intense fighting across the three cities that make up the nation's wider capital - Khartoum, Omdurman and Bahri - and smoke could be seen rising from several areas early on Monday.
"The neighborhood where we live in the center of Omdurman is looted publicly on a daily basis without anyone intervening to prevent it, with clashes and shelling continuing around us," said 37-year-old resident Mohamed Saleh.
In Khartoum East district, RSF troops who have spread out in neighborhoods across the capital were in full control and were looting extensively, said Waleed Adam, a resident of the area.
"You see them right in front of you, taking cars, money, gold - whatever they can get their hands on," he told Reuters by phone. "I guess it's just a matter of time until they come to my street."
The RSF says it has been working to protect civilians by arresting looters.
The war has also triggered unrest in Darfur in Sudan's far west, a region that was already suffering from mass displacement due to earlier conflict and where residents in several cities and towns have reported attacks by militias linked to Arab nomadic tribes.
On Monday, the RSF, which has its powerbase in Darfur and its origins in the Arab-dominated militias, released a video purporting to show that they had taken over the army headquarters in Kutum, a commercial hub and one of the larger towns in the state.
There was no immediate comment from the army, which had denied on Sunday that the RSF had taken the town.
There have been long communication blackouts in parts of Darfur, where aid groups have found it especially complicated to bring in new humanitarian supplies.
In El Obeid, a city 360 km southwest of Khartoum and on a key route from the capital to Darfur, residents reported large deployments of RSF forces and the closure of some roads.
Recent days have brought the first showers of the year in Khartoum, marking the start of a rainy season likely to complicate a relief effort already hampered by bureaucratic delays and logistical challenges.
In recent days at least 40 people were killed and dozens more were wounded in Kutum in North Darfur State, according to activists who monitor the region. Residents have also reported widespread looting and insecurity in the area.
Fighting between the army and the paramilitary Rapid Support Forces (RSF), who have been battling each other for more than seven weeks, intensified after the expiry late on Saturday of a ceasefire deal brokered by Saudi Arabia and the U.S.