Starting next month, unhoused people seeking state-provided Emergency Assistance shelter will receive services based on their circumstances and residential status, a dramatic change by the state long sought by critics of the Healey administration’s policies.

Gov. Maura Healey announced Tuesday that beginning August 1 the state will prioritize housing Massachusetts families who are homeless following a no-fault eviction, or as a result of a natural disaster, or any family that includes a veteran.

“I want to be clear to people, particularly outside of Massachusetts, who may have gotten word that this is a place to come, that we do not have room here in Massachusetts,” Healey said.

According to the governor’s office, families that include newborn children, or those with significant medical needs, or who may be at risk of domestic violence will also continue to receive priority.

Families who do not meet criteria for prioritization will still receive some housing support, but only on a limited basis.

Under the new rules, the state’s safety-net sites —  the overflow shelters for families who could not be housed elsewhere — in Chelsea, Lexington, Cambridge and Norfolk, will all be rebranded into “temporary respite centers” where families will be eligible to stay for five days.

In the time since the shelters filled, the Healey administration says they’ve worked to clear the way for immigrants to find jobs through negotiations with the federal government for work authorizations, provided English classes, and job placement, in part to give migrant families “the tools they need to move into stable housing and to meet the workforce needs of our employers.”

However, the governor said, the state has been at or near its 7,500 family shelter capacity since late last year and cannot maintain that position indefinitely.

“That’s why we are making changes to EA prioritization and transitioning our safety-net sites to five-day temporary respite centers. This is in line with the policies of other cities facing similar challenges as Massachusetts and will help give families some relief for a few days while they access the diversion services we can provide,” Healey said in a statement.

According to the MassGOP, Healey’s decision to change the shelter rules is in line with recommendations made by Republican state Senate Minority Leader Bruce Tarr and House Minority Leader Brad Jones.

“The Healey-Driscoll Administration is finally implementing some of the measures we’ve been advocating, despite repeated opposition from the Democratic supermajority. While it’s good to see some necessary steps being taken, it’s frustrating to consider the significant amount of money that has been wasted reaching this point,” party Chair Amy Carnevale said.

The cost of housing 7,500 families approaches $1 billion annually, according to state budget writers.

While the state’s conservative party welcomes Healey’s shelter policy changes, Carnevale said that it would have been better if she made the change months ago. Politics, the state Republican leader suggests, are the reason for the delay.

“These changes could have been enacted when Republicans first spoke out about the migrant crisis; however, the Healey-Driscoll Administration and the Democratic supermajority preferred to placate their base rather than address the crisis at hand. Their sudden shift now exposes a clear hypocrisy in their stance, preferring to play politics, rather than making the tough decisions for the betterment of the Commonwealth,” Carnevale said.

Healey, during a press conference Tuesday, said she was not aware of the Republican Party’s assertion and therefore could not respond.

House Speaker Ron Mariano said Healey’s decision to change the state’s shelter rules does not represent an abandonment of the state’s “right to shelter” laws, but instead reflects the realities of the situation. Massachusetts is alone among the 50 U.S. states in guaranteeing a right to shelter for families with small children or pregnant parents.

“The responses are measured. This works for all of us. We recognized that this was a limited program, we didn’t have unlimited resources. It was a necessary decision we had to make, and I certainly support the administration in this decision,” Mariano said. “It isn’t a step away from the right to shelter law, it’s a realization of the existence of limits to the law.”

Senate President Karen Spilka said that the immigration problem is not one that Massachusetts made for itself, but one that starts in Washington D.C.

“This is a federal issue. We still need federal help. We need federal immigration reform. We need federal funds. We need help, we cannot continue to do this alone,” she said.

The Healey administration announced Tuesday that stays at the state’s overflow emergency shelters, like the one at the former Norfolk prison, seen here, will be limited to 5 days under a new policy effective August 1. (Stuart Cahill/Boston Herald, File)