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Even as New York State prepared for a significant snowfall on Sunday, Gov. Kathy Hochul told New Yorkers that the state’s coronavirus forecast was improving:

“The Covid clouds are parting,” Ms. Hochul said.

For the past week, New Yorkers have watched with tentative optimism as the skyrocketing rate of new coronavirus cases began to slow, then fall. Test positivity rates are down as well, with just 13 percent of statewide tests on Saturday coming back positive compared to 23 percent a few weeks ago. In every region but one, seven-day averages of new cases were lower than the past three days.

This and other recent data show that the latest surge in New York driven by the Omicron variant may be starting to trend downward, and that several Northeast states — including New Jersey, Massachusetts, Connecticut and Rhode Island — may be heading in the same direction.

“Overall, the prognosis, the forecast for Covid is much brighter than it has been before,” Ms. Hochul said at a briefing, giving one of her clearest messages yet that the crush of cases caused by the Omicron variant might be beginning to lift. She went on: “That is very positive news, if our hospitalizations continue to go down as well.”

The caveat was an important one — while officials say that the number of people admitted to hospitals for Covid is slowing, hospitals remain under tremendous strain, struggling to manage staffing shortages that leave doctors and nurses no choice but to make difficult decisions about whose care to prioritize.

Although scientists believe Omicron can cause less severe disease than previous variants, the sheer number of cases has created a tsunami of patients seeking care. Cases remain extremely high in New York and elsewhere, and the United States is averaging a staggering 805,000 daily cases.

About half of all patients in hospitals across the city have Covid-19.

On Thursday, nurses working at overwhelmed city hospitals held a news conference and issued a plea for help. They shared stories of emergency rooms overrun and patients lying on stretchers in hallways as they wait for care — stories that could have been from the earliest days of the coronavirus pandemic.

Ms. Hochul has deployed the National Guard to help out in understaffed hospitals, but most of them have been directed upstate where the need was even greater. On Thursday, President Biden announced that two teams of military medical teams would be deployed to help New York City public hospitals. Ms. Hochul has requested an additional 800 clinical staff from the federal government to be deployed across the state as “strike teams,” according to her office.

Statewide, New York reported 51,264 new cases on Saturday, down from the peak of more than 90,000 a week earlier, according to a New York Times database. The average number of hospitalizations across the state appear to be leveling off as well.

Even so, the governor noted, there are many parts of the state where case numbers are high. She urged vigilance, and preparedness, once again asking people to get vaccinated and stay home if they felt ill.

She ended her briefing on a hopeful note. “Upstate is still not out of the woods yet, but it’s going to get there,” she said, adding: “Let’s do the right thing — get through this together.”