Major changes are coming next week to how homeless and migrant families access shelter services in Massachusetts.

Gov. Maura Healey on Tuesday announced plans to prioritize certain families for space in the state’s strained emergency shelter network, marking Beacon Hill’s latest effort to get a handle on continually climbing costs driven by unprecedented demand for shelter.

Starting Aug. 1, when the new rules take effect, families who are not among the prioritized groups will be eligible to stay for five days in state-run overflow shelter space in Chelsea, Lexington, Cambridge and Norfolk. State officials are now labeling those locations “temporary respite centers.”

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 told reporters Tuesday afternoon. “We are at capacity.”

Families who will be prioritized for shelter placement include those with veterans, newborns, and those who lose their housing because of a no-fault eviction or “sudden or unusual circumstances in Massachusetts beyond their control, such as a flood or fire,” Healey’s office said. People at risk of domestic violence or with significant medical needs are also on the priority list.

Other families who stay at a temporary respite center will then have to wait at least six months for a full shelter placement. They will be eligible for other aid, like help with travel expenses to other destinations and state programs that help with housing costs.

“This is the responsible approach,” Healey said. “It’s a humane approach. And it was important that we announced that today.”

While a 1980s Massachusetts law guarantees eligible homeless families the right to shelter, Healey and state lawmakers have implemented new restrictions in the past several months as the system stretched past its limits to accommodate both an influx of migrants and longtime residents struggling to afford housing.

Healey announced a 7,500-family cap on shelter capacity last fall, and the state Legislature in April passed a nine-month shelter stay limit, with some extensions available. Earlier this month, the state banned families in need of shelter from sleeping at Logan Airport.

Meanwhile, state officials have worked to help migrants staying in shelters get authorized to work in the U.S. and to connect shelter residents with jobs.

Republican state lawmakers have pushed for more shelter restrictions, including residency requirements. Democrats, who have supermajority control over both the House and Senate, have generally resisted those efforts.

“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,” Massachusetts Republican Party Chair Amy Carnevale said in a statement.

During debate earlier this year on the state budget, both the House and Senate rejected GOP-backed amendments seeking to prioritize homeless veterans for shelter space.

“We knew from the very beginning that we would have to have some ability to adjust this program going forward. And we put the decision-making in the hands of the administration because they were on the ground dealing with these folks as they come into Massachusetts,” House Speaker Ron Mariano said Tuesday. “They were the best to assess the situation, and the response is I think measured and I think this works for all of us. I mean, we recognize that this was a limited program. We didn’t have unlimited resources.”

The new time limit for overflow shelter does not apply to locations run by the United Way.

The Healey administration says it “will not be opening additional respite center sites due to operational and financial constraint.”

The newest overflow shelter, or temporary respite center, is located at a former state prison in Norfolk, and Healey’s office said that facility is approaching its capacity.

Norfolk Select Board Chair Jim Lehan said officials there learned about the shelter changes Tuesday morning, and that the move “dramatically changed” the implications for the town. Norfolk schools would have faced significant challenges in accommodating a wave of new students staying at the facility, he said.

“We’re all shocked, to be honest with you,” Lehan told GBH News. “This is such a turn around and so quick, so sudden.”

Marilyn Schairer contributed reporting.