Chris Adams, 36, has spent the past year of the pandemic living with his grandparents in Wichita, Kan., and being “extremely strict” about social distancing. “I never went out,” he said.But starting Monday, when all adults in Kansas become eligible for the coronavirus vaccine, Mr. Adams plans to find a vaccination site where there is an available appointment. “What I’m looking forward to is seeing my friends again,” he said.Kansas is one of six states — Louisiana, North Dakota, Ohio, Oklahoma and Texas are the others — that are expanding eligibility for the vaccine to all adults on Monday. Minnesota will follow on Tuesday, and Indiana on Wednesday.Gov. Laura Kelly of Kansas urged residents last week to seek out appointments, saying, “With the anticipated increase in supply from the federal government, we must get every dose of vaccine into arms quickly.”Even as vaccine eligibility continues to expand across America — nearly all states have pledged to make every adult eligible by May 1 — the United States has also reported an increase in new cases over the past week. About 75,000 new cases were reported on Friday, a significant increase from the 60,000 added the Friday before.States in the Northeast have accounted for about 30 percent of the nation’s new cases over the past two weeks, up from 20 percent in the first couple of weeks in February.In New York, there has been an average of 8,426 new cases a day, an 18 percent increase from the average two weeks earlier, according to a New York Times database. In New Jersey over the past week, there have been an average of 4,249 new cases reported daily, a 21 percent increase from the average two weeks earlier. And on Friday, Vermont set a single-day case record with 283 new infections; it is the first state to set a case record since Jan. 18.For many, the vaccine cannot come soon enough.Nicole Drum, 42, a writer in the Kansas City, Kan., metro area, cried on Friday when she found out that she would be eligible to get the vaccine as early as Monday. She started calling pharmacies and looking online for available appointments “within minutes of the news breaking,” she said.Ms. Drum called about 10 places without success. She had more luck on a county website, and booked an appointment for Wednesday.She said she planned to wear a special T-shirt saying “I believe in science” to her appointment. “I got myself a fun I’m-getting-the-vaccine outfit,” she said, laughing.She also plans to take her 4-year-old son with her, because she wants him to see “how research and science and people coming together can really help stem these kinds of things,” she said.“I want him to know that there’s no need to be afraid all the time of big scary things, because there are always helpers trying to figure this out,” Ms. Drum said. “While the solution might be something that’s a jab in the arm that hurts a little bit, it’s worth it.”See How the Vaccine Rollout Is Going in Your County and StateSee where doses have gone, and who is eligible for a shot in each state.