Nearly 300,000 people in the region had received their first dose of the vaccine as of Monday, and about 22,000 people were expected to do so in the coming weeks. “I am encouraged,” Dr. Melendez said. “Do we have a long way to go? Absolutely. But we have come a long way.”

Calls for Monday’s appointments in Louisiana began three weeks ago, after Gov. John Bel Edwards announced the state’s expansion of eligibility beyond those with underlying health conditions. But Ruston Henry, a pharmacist in New Orleans, noted that his customer base had grown only slightly younger. “We’re seeing old to middle-aged people,” he said. “We’re not seeing many people who are ages 16 to 20.”

As of Thursday, about 15 percent of Louisianans, or about nearly 721,000, had been fully vaccinated, while more than 1.1 million have received at least one dose, officials said.

In Ohio, the demand for vaccines has been palpable, with appointments filling up quickly and some people putting their names on more than one waiting list. In Franklin County, appointments for the entire week were filled within two hours of being made available. Tan Thai, who works at the Old Village Pharmacy in Columbus, said appointments for her most recent batch of vaccines were snapped up in 10 minutes.

“People are so desperate right now,” Ms. Thai said. “They are looking everywhere.”

Dr. Varon, who has witnessed firsthand the devastation brought by the pandemic in the Houston area, said widespread vaccination efforts did not arrive soon enough. More than 48,000 people have died from Covid-19 in Texas, and he fears a surge after spring break and Easter.

“We are expecting a fourth wave,” he said. “It’s going to happen. Things are not going to be good over the next few weeks.”

He cautioned people from letting their guard down until a majority of the population has been fully vaccinated and herd immunity is reached, a number that would assure that medical systems are not overwhelmed with patients. “We are not even close,” he said of the vaccination rates.

He especially urged communities of color — some of which have shown apprehension to sign up for a virus vaccine — to listen to their doctors and public officials and go to the nearest site. “I tell you, please, the vaccines are safe,” Dr. Varon said. “Get vaccinated. This is your only chance to survive this pandemic.”

Reporting was contributed by A. Colleen DeGuzman from McAllen, Texas, Marina Trahan Martinez from Dallas, Katy Reckdahl from New Orleans, and Lucia Walinchus from Columbus, Ohio.

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