A key demand of the opposition movement has been the removal of Article 8 of the constitution which stipulates that the Baath party is the sole "leader of state and society."
But in defiance of growing international condemnation, hundreds of Syrian security services raided homes in the port city of Latakia on Wednesday, activists said.
The Britain-based-based Syrian Observatory for Human Rights said more than 700 members of the security services took part in the operation in the southern district of Ramel, arresting people on lists.
"Heavy gunfire continued in most opposition neighbourhoods" overnight, the Britain-based group said.
In Jabal al-Zawya, a village in Idlib province near the border with Turkey, security forces shot dead a man standing on his balcony, it said.
In Homs, a sniper shot dead a civilian in the Armenian neighbourhood of the city in central Syria, and scores of people were arrested across provincial towns, the Observatory said.
It said security forces in Damascus carried out dawn raids in Rukn Eddin district, where electricity was cut off, and arrested dozens of activists. Dozens of others were arrested overnight on the outskirts of the capital.
Western countries are to call on the United Nations top human rights body to hold a special session on the deteriorating rights situation in Syria, diplomats in Geneva said.
On Tuesday, British Foreign Secretary William Hague stepped up the pressure and warned that Assad was "fast losing the last shreds of his legitimacy."
And US Secretary of State Hillary Clinton urged Arab heavyweight Saudi Arabia and Syria's neighbour Turkey to push Assad to step down.
Switzerland on Wednesday widened its sanctions against the Assad regime, adding 12 individuals to a list of key players under financial embargo and travel restriction.
But the head of Russia's arms export agency, cited by the Interfax news agency, said Moscow was continuing to supply weapons to its traditional ally Damascus.
"While no sanctions are announced, while there are no orders or directions from the government, we are obliged to fulfil our contractual obligations, which we are now doing," Rosoboronexport chief Anatoly Isaikin said.
Since Sunday, 30 civilians have been killed in Latakia in a military offensive during which gunboats went into action for the first time since the start of pro-democracy revolts in mid-March, according to activists.
The official news agency SANA has denied any maritime operation and on Tuesday quoted a military official saying security forces were "hunting armed men" in Latakia districts "who opened fire on residents."
The UN Relief and Works Agency for Palestinian refugees reported that more than half of the 10,000 refugees of Ramel camp in southern Latakia had fled under fire from Latakia districts.
Syrian Foreign Minister Walid Muallem accused countries he did not identify of "putting pressure on Syria to stop the violence but ignoring that terrorist armed groups are behind this violence," SANA reported.
An AFP journalist on a government tour of the eastern oil hub of Deir Ezzor said dozens of army vehicles left the city on Tuesday after a 10-day operation, which activists said killed more than 30 people.
Hours later the Syrian Observatory reported that one person was killed when security forces opened fired to disperse an anti-regime protest where "hundreds of people" marched in the city's Takaya Street.
Rights groups say the crackdown has killed 1,827 civilians since mid-March, while 416 security forces have also died.
The Wall Street Journal reported on Wednesday that Syrian diplomats were intimidating expatriates who speak out against the regime, and reporting back home where dissidents' relatives are then threatened and arrested.
The Obama administration told the Journal it had "credible" evidence that the regime is using the reports to target relatives of those living overseas, particularly Syrian-Americans who have joined peaceful US protests.
Embassy staffers were tracking and photographing protesters, it said.