Sen. Bernard Sanders said Sunday that Hillary Clinton must cut ties with the Clinton Foundation, adding more pressure to the chorus of critics who say the former State Department secretary’s entanglements are damaging her.

The comments of Mr. Sanders, who fought a grueling primary battle with Mrs. Clinton this year, come just two days after the FBI released its notes of its criminal investigation interview with the former top diplomat, revealing that, despite being one of the country’s few officials able to deem material classified, she showed little understanding of that weighty responsibility.

Both her behavior toward her family’s foundation and her use of the secret email system during her time at the State Department — which some critics say was intended to further hide the foundation’s close connections during her time in office — have dogged Mrs. Clinton for months, and have dented her standing with voters.

Mr. Sanders, speaking on NBC’s “Meet the Press” program, said he wasn’t sure whether the foundation needs to be shut down altogether, but said if Mrs. Clinton wins the White House, she needs to divorce herself from it.

“She should not be involved at the very least,” he said — and said it was possible the entire operation should be shuttered, though he acknowledged that given the charitable work it does, he’s not sure.

New emails retrieved from one of Mrs. Clinton’s aides showed top foundation executives regularly sought special favors for themselves and their supporters from the State Department.

The emails have also raised even tricker questions about Mrs. Clinton’s ability to handle classified information, after investigators found she used her easily penetrated server kept at her New York home to send or receive more than 100 email chains that included classified information.

FBI Director James B. Comey earlier this summer closed a yearlong investigation by saying he didn’t think she should be charged under a law that punished negligence in handling of classified material. Mr. Comey said it was clear Mrs. Clinton was negligent, but he said she was so inept at her understanding of both classified information and technology that he doubted a prosecutor could prove she knew the risks she was taking with national security.

On Friday the FBI released a small portion of its investigative files, including its notes of agents’ interviews with Mrs. Clinton, and the staggering findings revealed that the former first lady, senator and top diplomat didn’t know how classification worked.

“Clinton could not give an example of how classification of a document was determined,” the investigators said in the notes of their interview.

In her interview with the FBI, Mrs. Clinton repeatedly said she relied on others at the State Department to handle that area of her work, and if they handled information without marking it, she assumed it wasn’t classified.

And when asked specifically about emails with paragraphs marked with a “(C),” designating classified information, Mrs. Clinton said she thought they denoted an alphabetical ordering, not secrets.

Some of the information Mrs. Clinton handled by email included potential drone strike targets. She said she thought those conversations were “part of the routine deliberation” and didn’t seem to require any special extra care when it came to classification, the FBI agents said in their notes of their interview.

The interview also continued to pose problems for Mrs. Clinton’s public explanation of her email use.

The former secretary of state at first publicly and emphatically insisted that she never received or sent any classified emails, before changing her public words to “classified at the time.” The FBI found even that statement to be untrue.

Agents also said she had more than a dozen BlackBerry devices and five iPads during her time as secretary, conflicting with her initial public explanation that she only set up the private server because she wanted to use a single device.

But while the FBI found that Mrs. Clinton had used 13 mobile devices, she could not produce any of them. One staffer said he had destroyed two of them, one by hitting it with a hammer.

Republican vice presidential nominee Mike Pence, the governor of Indiana, said the FBI revelations showed a disturbing pattern.

“What’s evident from the notes, what’s evident from all of the revelations over the last several weeks, is that Hillary Clinton operated in such a way [as] to keep her emails and, particularly, her interactions, while secretary of state, with the Clinton Foundation, out of the public reach, out of public accountability,” he said on “Meet the Press.”

On ABC’s “This Week” program, Mr. Pence went further into the annals of American political corruption, calling the FBI report “more evidence that Hillary Clinton is the most dishonest candidate for president of the United States since Richard Nixon.”

But Democratic vice presidential nominee Tim Kaine said the fact that the FBI’s investigative files are being made public is proof that Mrs. Clinton wants transparency.

“She did make a mistake … by deciding she wanted to use one device rather than multiple devices. She’s apologized for that. She said it was a mistake. And she’s learned from it,” he said. “But these notes, which Hillary urged be made public, demonstrate clearly why the FBI saw no need for additional criminal proceedings.”

Mrs. Clinton blamed her mishandling of classified information on her staffers, saying she trusted them to know what was classified and what wasn’t. She said if the information wasn’t clearly marked, it didn’t strike her to question their decisions.

Democrats continued in that vein Sunday, blaming Mrs. Clinton’s confusion on mistakes or changes made by others.

Labor Secretary Thomas E. Perez said Sunday that the FBI report showed that “there were a lot of markings that were at times incorrect and ambiguous, and weren’t in keeping with what the Department of Justice would have in place when they were marking documents that are confidential.”

“A lot of these documents were retroactively reclassified, so the classification system presented challenges as well,” Mr. Perez said on CNN’s “State of the Union.” “So again, I think this has been investigated very, very thoroughly, and she understands that she shouldn’t have done it, and I think she’s learned a lot from that.”

Mr. Kaine argued Sunday that some emails were mislabeled.

“My recollection of the [FBI director] Jim Comey testimony before Congress was that many — that there were emails that contained classified information that had been improperly marked,” Mr. Kaine said on ABC’s “This Week.”

“So when she received the email, the material that was classified, which is supposed to be flagged and identified as classified, in many instances was improperly labeled,” said the Virginia senator.

Based on his experience with handling sensitive material in the Senate, he said it’s sometimes hard to determine whether a document is classified or isn’t.

“Now I’m on two committees in the Senate, the Armed Services and Foreign Relations committees, where we do look at classified material,” Mr. Kaine said. “But we look at so much material, unless it is specifically pulled out and identified, it is difficult to know sometimes whether a statement or a paragraph is classified or not. And that’s what she was saying.”

He added that, “Unless it is identified in the way that it should be, it is difficult to know whether a particular material is classified or not.”

Rep. Elijah E. Cummings, the ranking Democrat on the House Oversight Committee, said the FBI should have released an email exchange between Mrs. Clinton and former Secretary of State Colin Powell where Mr. Powell warned that if Mrs. Clinton used a BlackBerry to do official business, the messages would be subject to open records laws.

Mrs. Clinton told the FBI that Mr. Powell’s advice didn’t factor into her decision-making.

Republicans, meanwhile, said the FBI should have released documents from one of the companies that operated the server kept at Mrs. Clinton’s New York home that handled her email traffic.

Rep. Trey Gowdy, who led the Benghazi probe that forced the revelation of the emails, said the timeline of Platte River Networks’ involvement in the server would be “instructive” for voters.