By Christina Maxouris and Holly Yan, CNN

(CNN) -- At first, the news sounded great: Fully vaccinated Americans don't need to wear masks indoors, with just some exceptions, the US Centers for Disease Control and Prevention said.

But most of the country is not fully vaccinated. And as more businesses drop their mask mandates, it's impossible to tell who really is.

So while the CDC's new guidance is scientifically sound, the messaging was botched, former US Surgeon General Jerome Adams said.

"I think that the play call was right, but they fumbled the ball at the 1-yard line in terms of communicating this to the public," said Adams, who served during the Trump administration.

Many public health officials told him "they are upset, they were blindsided by this new information," Adams said.

"It was a little bit of whiplash for the American public in terms of them saying just a week before, 'Keep your mask on,' and then all of a sudden they're saying now you can take them off," Adams said.

Emergency physician Dr. Leana Wen said the whiplash was particularly drastic because she said the CDC had been too careful in its guidance on what vaccinated people can do. In April, the CDC said fully vaccinated people can unmask at small outdoor gatherings and when dining outside with friends from multiple households.

Now, Wen said, the pendulum has swung in the opposite direction. The new guidance could essentially allow anyone to go mask-free indoors without proof of vaccination. And that kind of honor system might jeopardize others, including children under the age of 12 who can't get vaccinated yet and adults who might be at more risk than they think.

"We've seen governors and mayors and business owners drop mask mandates, and as a result of that, we've now made life much less safe for people who are unvaccinated, for immunocompromised individuals and for young children who cannot yet be vaccinated," Wen said.

"I think the CDC meant to say something really good, which is these vaccines are really protective. There were unintended consequences of their actions."

Why fully vaccinated doctors will keep wearing masks

Adams said his two teenage sons have been vaccinated, but his 11-year-old daughter is not yet eligible.

"So when we go out in public, even though I don't have to wear a mask anymore because I've been vaccinated, I wear one with I'm with my daughter because I don't want her to feel like she's the only one left out," the Trump administration surgeon general said.

"We're in this together," he said. "Sometimes, you do things even if you don't feel like they'll benefit you because they'll benefit others. We don't want our little ones to be stigmatized or at risk."

Adams said he'll also keep masking up "in those public settings where there may be a large number of people who aren't vaccinated."

As of Monday, about 37% of Americans have been fully vaccinated, according to CDC data.

One easy way to avoid a questionable honor system is "to know who's vaccinated and who's not vaccinated, and the unvaccinated folks would still be required to wear a mask," said Dr. Jonathan Reiner, a professor of medicine and surgery at George Washington University.

But unfortunately, "this is where politics comes in."

"If we had had a very simple electronic system from the beginning, basically you have a pass on your phone that turns green after you've been vaccinated, businesses and venues all over the country would be able to say, 'Good news, if you have a green pass, you don't need to wear a mask,'" Reiner said.

"We don't have that."

'Not everybody has to rip off their mask' CDC Director Dr. Rochelle Walensky said the agency's new guidelines don't mean that every American who's been vaccinated should take their masks off.

For example, Walensky said immunocompromised people and others at higher risk should consult with their physicians before deciding to stop wearing a mask.

"Not everybody has to rip off their mask because our guidance changed," she said. "If you are concerned, please do consult your physician before you take off your mask."

Walensky said different parts of the US have different vaccination and Covid-19 rates, and fully vaccinated people should take that into account deciding whether to mask up or not.

She said the CDC is asking businesses to make sure they help their employees get vaccinated.

And schools should keep their mask rules and other safety measures for at least the rest of this school year because many students will not be fully vaccinated in the next couple months, the CDC said.

The Food and Drug Administration expanded the emergency use authorization for Pfizer's Covid-19 vaccine last week to include children 12 to 15 years old. Children under the age of 12 are not yet eligible to receive a Covid-19 vaccine in the US.