This Texas city has a relatively high vaccination rate, but it's still fighting to get shots in arms amid rising cases

Overall, Travis County, which includes Austin, is doing relatively well in terms of vaccinations. About 63.4% of the county's population over age 12 is fully vaccinated, according to data Monday from the Texas Department of State Health Services, compared with about 52.9% of the state's population over age 12.

But it's not enough to stem the rising tide of infections and hospitalizations driven by the Delta variant, officials say.

Austin Mayor Steve Adler told CNN the county and the city had done well, and he touted the high rate of vaccinations. "Which just goes to show this Covid variant, Delta, is just that much more infectious and having that much greater impact," he said.

Early on, thousands of people a day were getting vaccinated in Austin, Dr. Desmar Walkes, Austin-Travis County health authority, told CNN. Vaccines were in such high demand that some people initially had to be turned away.

Nowadays? "We're looking maybe at 50 to 100 people depending on how many strike teams we have out on any given day," Walkes said.

"Our ICU capacity is reaching a critical point where the level of risk to the entire community has significantly increased, and not just to those who are needing treatment for COVID," Walkes said Friday in a statement. "If we fail to come together as a community now, we jeopardize the lives of loved ones who might need critical care."

Added Adler: "It is beyond frustrating that it's so hard to get vaccines in people's arms today when we were giving thousands of them out just a couple months ago. And we're doing everything we can."

Now, the "labor intensive effort" involves "virtually going door-to-door," he said.

"We're trying to find people where they are," Adler said. "We're working through trusted voices and communities, working with churches and faith institutions, faith leaders."

Camila admits she was nervous about getting a shot. She says she initially didn't want to get vaccinated. But she was the last member of her family to need it -- everyone else had been vaccinated.

So, she decided to listen to someone she trusts -- her mom, who told her the vaccine was the best path to a return to normalcy.

"My mom was talking to me in the car, and she was like, 'You know, if you want the world to get better, we need to help,'" Camila said. "Everyone needs to get vaccinated."

After weeks of progress, Covid-19 cases are rising rapidly in states across the country. In Texas, the seven-day moving average was 9,789 new daily cases on Monday, according to a CNN analysis of Johns Hopkins University data. That's way up from July 1, when the seven-day moving average of new daily cases was about 1,500.

Austin is also feeling the surge. The city's Covid-19 dashboard showed a seven-day moving average of about 365 new daily cases on Monday, up from about 34 on July 1. Hospitalizations are also up, with a seven-day average of 346 patients reported over the prior week as of Monday, compared to 57 patients at the beginning of July.

"This Delta variant has really caused an alarming increase in the number of cases," Walkes told CNN. "We've gone from 30 cases a day to almost 400 cases a day in a matter of almost two-and-a-half, three weeks."

"That's because we have still a lot of people who are unvaccinated," she said.

Nearly everyone in the city's hospitals aren't vaccinated, Adler told CNN.

"Almost everyone in our ICUs are people who are not vaccinated. We have no one on ventilators in our city that are vaccinated," he said. "This is an epidemic among the unvaccinated."

Last week, the city moved to Stage 4 out of 5 of its risk-based guidelines, recommending that partially or unvaccinated people avoid private gatherings, dining, travel and shopping unless it's absolutely essential. And everyone -- vaccinated or not -- is advised to wear a mask.

Meantime, Gov. Greg Abbott has said he will not impose a statewide mask mandate and has previously banned local government entities from requiring vaccines.

Texans know "what the standards are, what practices they want to adopt to help protect themselves," he told CNN affiliate KPRC last week. "This is time for individual responsibility."

But on the front lines, Adler said, that sort of messaging is "making it harder for us to get the behaviors we want, when our governor is not ready to actually join in a way that sends an unambiguous message to the community about the need to get vaccinated and about the ability and importance for people to wear masks while infection levels are real high."

Still, Austin officials and the governor agree that vaccines are effective against the virus.

Health officials are focused now on what Walkes describes as the "movable middle" -- people who are looking for more information or need their concerns to be addressed on a one-on-one basis.

So-called strike teams are mobilizing in communities to find these individuals where they live, work and play, Walkes says.

But that means there's more manpower being spent to vaccinate fewer people. And officials also have to fight misinformation, convincing people the vaccines are safe and effective, she says.

"Every person that we can vaccinate is another person that's not going to be sick," she said. "The vaccines work to prevent severe illness, hospitalization and death."

The efforts Austin has made to increase vaccinations is a pattern playing out across the state.

The Texas Department of State Health Services will award $10 million to local organizations promoting vaccines, like educational agencies, faith-based organizations, community coalitions and nonprofits, it has announced.

There has been some success in Austin, Walkes says, describing a "slight uptick" in people coming out to get vaccinated, especially when it comes to kids like Camila.

"Particularly, we're seeing parents bringing kids out to get vaccinated," Walkes said, "because we're weeks away from reopening school, and we want to do what we can to keep our kids safe."