Ethiopia says it had captured or killed most commanders of a rebellious northern force, while Tigray's fugitive local leader says occupying soldiers are engaging in looting, sparking civilian protests.
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Neither side gave proof for their assertions about the month-long war in the mountainous region bordering Eritrea.
Phone communications have been down and access severely restricted.
Fighting between Prime Minister Abiy Ahmed's federal army and forces loyal to the region's former ruling party, the Tigray People's Liberation Front (TPLF), erupted on November 4.
Thousands of people are believed to have died while more than 45,000 refugees have crossed to neighbouring Sudan.
TPLF leaders, who enjoy strong popular support in Tigray, appear to have fled to surrounding mountains and say they have begun a guerrilla-style resistance.
TPLF leader Debretsion Gebremichael, one of the most wanted men in Ethiopia, told Reuters there were protests in Mekelle, which is home to 500,000 people, due to looting by Eritrean soldiers.
"Eritrean soldiers are everywhere," he said, repeating an accusation that President Isaias Afwerki has sent soldiers over the border to back Abiy against their mutual foe.
Both Ethiopia and Eritrea have denied that.
Ethiopia says the TPLF wants to internationalise the conflict as a way to force the government, which appears to hold all the major towns, into international mediation.
Debretsion, a 57-year-old former guerrilla radio operator, gave no evidence of looting or the presence of Eritreans.
A senior military commander, General Tesfaye Ayalew, said "almost all of the enemy", including former federal colonels and generals who fought on the Tigrayan side, were defeated or dead.
"But the ones who made the plans and the criminals are still on the run and hiding," he told state-affiliated Fana TV.
Relief agencies are extremely worried about lack of food, fuel, medicines and even bodybags in Tigray.
Convoys are on standby to take aid in.
In Sudan, refugees recounted horror stories of fleeing from Tigray along roads strewn with bodies and also spoke of Eritrean involvement.
Tewodros Tesera, a surgeon from the border town of Humera, said shells had come from the Eritrean side over the Tekeze River in the early days of battles.
"The shells were falling in front of the hospital where I work," he recalled by phone from the Hamdayet refugee camp.
"We heard 'boom', then 20 seconds, then whistle, then a second heavy bombing, which struck the ground and buildings. Houses, a mosque and a church were damaged."
Australian Associated Press
