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While all adults in Virginia are now eligible to make coronavirus vaccine appointments, supply constraints mean it could be weeks before residents in Fairfax and other areas with high demand are able to make appointments, officials said Monday.

An additional 3 million people became eligible for the vaccines on Sunday in Virginia, with Gov. Ralph Northam (D) saying the state is on track to get first doses to all adults who want them by mid- to late May.

Northam was in Tysons Corner on Monday in advance of the opening of a mass vaccination clinic at a former Lord & Taylor department store that is ready to administer 3,000 shots daily six days a week in hopes of eliminating the backlog in Fairfax County. The clinic will open Tuesday by appointment only.

“The sooner we can get people vaccinated the better and the sooner we can get to herd immunity,” Northam said at a news conference.

While the state on Sunday had moved to Phase 2, which includes all adults 16 and older regardless of age, health condition or employment, Fairfax County public health officials were still working through lists of people who preregistered for the vaccine in Phases 1a through 1c, which includes people older than 65, those with underlying health conditions and essential workers.

Jeff C. McKay (D-At Large), chairman of Fairfax’s Board of Supervisors, who for weeks has said the county does not have enough vaccine supply to meet the demand, urged residents to remain patient.

“We know that demand is outstripping supply. It really has since the beginning,” he said at a news conference at the Tysons Corner clinic Monday.

While those 16 and older can still register for appointments, McKay said, “We want to make sure we get those Phase 1 people who have been in our queue waiting to be vaccinated before we start opening it up to Phase 2 people at the Fairfax County health control clinics.”

The wait has led to frustration for residents such as Marilyn Adams, 65, of McLean, who said as of Monday afternoon that she could not find a pharmacy appointment for her son, a 23-year-old college student in Florida, for when he comes home for the summer in a few weeks.

“It’s misleading. It’s completely inaccurate,” said Adams, who searched within a 25-mile radius of her home in Fairfax County. “It gave me small towns in Virginia I have never heard of, but I was willing to drive there.”

She worried young people who are unwilling to continue checking websites for available appointments will get frustrated and give up.

Other Northern Virginia localities have differing rules. Arlington officials on Friday began inviting people registered under Phase 2 to book appointments and will continue to prioritize Phase 1 people who sign up now, county health district spokeswoman Jessica Baxter said.

Public health officials in Fairfax and Arlington said residents registered under Phase 2 who did not want to wait for the state to invite them to make appointments should frequently check for appointments at pharmacies listed on the federal website VaccineFinder.org.

Although all of the state is technically in Phase 2, pharmacy appointments fill up quickly and health departments are inviting people who preregistered to make appointments based on limited supply. Danny Avula, the state’s vaccine coordinator, said Monday that Pfizer and Moderna supplies are expected to remain flat for a few weeks.

Another obstacle for Northern Virginia’s large immigrant populations is the fact that the federal website to make appointments is only in English.

Avula said Virginia will ask the Centers for Disease Control and Prevention, which developed the site, if it plans to include other languages. Non-English speakers can still make appointments through vaccinate.virginia.gov, the state health department’s website, he said.

The new federally funded mass clinic can accommodate multiple languages, with access to remote interpreters available near the facility’s reception area. Northam on Monday toured the former store, where boxes of hand sanitizer, masks and Band-Aids replaced Ralph Lauren clothes and other goods in the cavernous space that symbolized the pandemic’s devastating impacts on the state’s economy.

In Virginia, 39.9 percent of residents — or 3.4 million people — had received at least one dose as of Monday, according to state health department data. Maryland is a close second with 39.9 percent of residents with one dose as of Monday.

In D.C., about 30 percent of residents received at least one dose as of Friday, according to D.C. Health data.

New infections in D.C. remain relatively flat. The seven-day average of new cases per 100,000 was 17 on Monday, which is up slightly from one month ago. The District reported 78 new cases and zero deaths on Monday.

In Maryland, the seven day average of new cases per 100,000 was 21 on Monday, which is up from 15 a month ago. The state on Monday reported 631 new cases and 12 deaths.

In Virginia, the seven-day average of new cases per 100,000 was 17 on Monday, which remains flat from one month ago. The state on Monday reported 978 new cases — the first day the number dipped below 1,000 in months — and 14 deaths.

D.C. Mayor Muriel E. Bowser (D) announced more steps toward normal in the city on Monday, plotting a course for a government that will resume pre-pandemic operations over the months to come.

Deanwood and Southeast, two of D.C.’s libraries that have been closed since last March, will reopen May 3, and the renovated Southwest library will open May 6. All of the library branches will end their 2 p.m. to 3 p.m. closed-for-cleaning hour that has been the norm during the pandemic, and many branches will resume Saturday hours for the first time since last spring.

At least one library in each ward will also take on a new role starting Monday: do-it-yourself coronavirus testing sites. Residents can pick up a sealed test kit at the library, swab themselves and leave it for testing.

The health department announced a change in policy on Monday, saying that while the city will continue to make a small portion of its vaccine doses through its city registration website available to non-D.C. residents who work in essential jobs in the District, the department had ordered D.C. hospitals to stop giving shots to nonresident workers.

Bowser pointed out that with the general public eligible for vaccine doses across the region, those workers can now seek vaccine doses in their home states.

Julie Zauzmer contributed to this report.