For the residents of embattled eastern Ukraine, things are looking dicier by the day. That is on the ground. Elsewhere in Ukraine, what should be summer silence overhead is increasingly punctured by air alerts and sometimes the sounds of Russian missiles.

The governor of the last remaining eastern province partly under Ukraine’s control urged his more than 350,000 residents yesterday to flee as Russia escalated its offensive and air alerts were issued across nearly the entire country.

Governor Pavlo Kyrylenko said getting people out of Donetsk province is necessary to save lives and enable the Ukrainian army to better defend against the Russian advance.

“The destiny of the whole country will be decided by the Donetsk region,” Mr. Kyrylenko told reporters in Kramatrosk, the province’s administrative center and home to the Ukrainian military’s regional headquarters.

“Once there are less people, we will be able to concentrate more on our enemy and perform our main tasks,” he said.

The governor’s call for residents to leave appeared to represent one of the biggest suggested evacuations of the war, though it’s unclear whether people will be willing and safely able to flee. According to the UN refugee agency, more than 7.1 million Ukrainians are estimated to be displaced within Ukraine, and more than 4.8 million refugees have left the country since Russia’s invasion started February 24.

President Zelensky said air alerts were issued Tuesday night in nearly all areas of the country, in many places after long periods of relative calm.

“You should not look for logic in the actions of terrorists,” Mr. Zelensky said in his nightly video address. “The Russian army does not take any breaks. It has one task — to take people’s lives, to intimidate people — so that even a few days without an air alarm already feel like part of the terror.”

Ukraine’s military is of course not taking any breaks either. The Ukrainian Air Force said that overnight Tuesday it destroyed six cruise missiles over the Dnipropetrovsk region and three over western Ukraine.

Much of the current military activity is concentrated in Ukraine’s east. The Kramatorsk governor said that because they house critical infrastructure such as water filtration plants, Russia’s main targets are now his city and a city 10 miles to the north, Sloviansk. Mr. Kyrylenko described the shelling as “very chaotic” and lacking “a specific target … only to destroy civilian infrastructure and residential areas.”

Sloviansk also came under sustained bombardment Tuesday. Mayor Vadim Lyakh said on Facebook that “massive shelling” pummeled Sloviansk, which had a population of about 107,000 before Russia invaded Ukraine more than four months ago. The mayor, who urged residents hours earlier to evacuate, advised them to take cover in shelters.

At least one person was killed and seven were wounded Tuesday, Mr. Lyakh said. He said the city’s central market and several districts came under attack, adding that authorities were assessing the extent of the damage.

The barrage targeting Sloviansk indicated Russian forces were advancing farther into Ukraine’s Donbas region, a mostly Russian-speaking industrial area where the country’s most experienced soldiers are concentrated.

Sloviansk has previously taken rocket and artillery fire during Russia’s war in Ukraine, but the bombardment picked up in recent days after Moscow took the last major city in neighboring Luhansk province, Mr. Lyakh said.

The Ukrainian military withdrew its troops Sunday from the city of Lysychansk to keep them from being surrounded. Russia’s defense minister and President Putin said the city’s subsequent capture put Moscow in control of all of Luhansk, one of two provinces that make up the Donbas, but the regional governor said Tuesday that fighting was continuing on Lysychansk’s outskirts. He said Russian forces were moving weaponry to Donetsk.

The question now is whether Russia can muster enough strength to complete its seizure of the Donbas by taking Donetsk province, too. Mr. Putin acknowledged Monday that Russian troops who fought in Luhansk need to “take some rest and beef up their combat capability.”

Meanwhile, Moscow-installed officials in Ukraine’s southern Kherson region on Tuesday announced the formation of a new regional government, with a former Russian official at the helm.

The head of the new Moscow-backed government in Kherson, Sergei Yeliseyev, is a former deputy prime minister of Russia’s western exclave of Kaliningrad and also used to work at Russia’s Federal Security Service, according to media reports.

It wasn’t immediately clear what would become of the “military-civic administration” the Kremlin installed earlier. The administration’s head, Vladimir Saldo, said in a Telegram statement that the new government was “not a temporary, not a military, not some kind of interim administration, but a proper governing body.”

Kherson’s Russia-installed administration previously stated plans for the region to become part of Russia, either through a referendum or other means.

Russia’s overall aims with respect to Ukraine appear to be little changed since the invasion began. Russia’s Security Council secretary, Nikolai Patrushev, said as much yesterday. According to the Institute for the Study of War, Mr. Patrushev said that the Russian “military operation” in Ukraine will continue until Russia achieves its goals of protecting civilians from “genocide,” “denazifying” and demilitarizing Ukraine, and obliging Ukraine to be permanently neutral between Russia and NATO — almost exactly restating the goals Mr. Putin announced in his February 24 speech justifying the war.