The State Department said Tuesday that former Secretary Hillary Rodham Clinton never signed the document affirming that she turned in all classified information when she left in 2013, as the legal jeopardy to the Obama administration continued to grow over her treatment of emails.

House Speaker John A. Boehner, the Ohio Republican overseeing several congressional investigations into Mrs. Clinton’s emails, said an independent third party needs to take control of her email server in order to clear up remaining questions.

Judicial Watch went to court this week to ask for a hearing on whether the State Department broke the law in failing to search Mrs. Clinton’s emails for records related to the administration’s 2012 Benghazi talking points — part of a pattern that likely has spoiled dozens of open-records requests related to Mrs. Clinton’s tenure.

“The way forward is for the secretary to turn over all of her emails that pertain to the public,” Mr. Boehner said. “But some neutral third party is going to have to make some decision about what documents are, quote, ‘personal’ and which ones are public records. And thus far, she’s been unwilling to do this.”

Mrs. Clinton acknowledged this month that she did not use a State Department-issued email account to conduct her government business, but instead set up a server kept at her house in New York. Nearly two years after she left office, she turned over more than 30,000 emails to the State Department.

She said she believes she complied with the law and that she did not keep some 32,000 other emails she deemed private. She refused to turn over her server to a third party.

The White House, hoping to keep the issue at arm’s length, has pushed questions about Mrs. Clinton’s behavior to the State Department, which has in turn pushed reporters to ask Mrs. Clinton about her behavior.

After days of prodding, though, State Department spokeswoman Jen Psaki confirmed Tuesday that the department does not have any record of Mrs. Clinton’s completion of a standard form, OF-109, certifying that she had turned over all classified information in her possession.

Ms. Psaki also said the State Department could not find the OF-109 forms for Mrs. Clinton’s two predecessors under President George W. Bush: Colin L. Powell and Condoleezza Rice. Ms. Psaki said signing the form wasn’t a legal requirement as much as a best practice.

She signaled that Mrs. Clinton and other secretaries are held to different standards than most other employees.

“There’s a difference between also secretaries of state or former secretaries and staff at lower levels,” she said.

Congress and the courts are likely to remain interested in Mrs. Clinton’s emails for the foreseeable future, even as she finalizes plans to run for the White House.

Mr. Boehner said the special investigative committee looking into the Benghazi attack needs to see related emails, and the House Oversight and Government Reform Committee will continue to look at open-records laws and how they are working.

He said Mrs. Clinton’s questionable email behavior would not have been spotted but for the Benghazi investigation. The committee requested communications about the 2012 terrorist attack on a diplomatic compound in Libya and discovered a deficit of emails from Mrs. Clinton — though committee members spotted several messages using an account that was not issued by the government.

The State Department then requested emails from all previous secretaries of state, and Mrs. Clinton turned over hers in December.

She has said she used a nongovernment account because it was more convenient but believed she was following the law because she generally tried to email others at their official accounts, thinking that would ensure the messages were retained as required by law.

Open-records advocates have poked holes her argument, saying requests seeking communications from Mrs. Clinton wouldn’t necessarily include searches of lower-level employees’ emails.

Judicial Watch President Tom Fitton said his group has 160 open-records requests that could have been at least partially spoiled by Mrs. Clinton’s decision to keep her emails private.

He said the State Department for years has responded to requests for documents, including Mrs. Clinton’s communications, by saying it has searched everything it has — but never mentioned that Mrs. Clinton’s emails were not part of that.

Judicial Watch argued in federal court this week that the State Department has been misleading open-records advocates and federal courts by claiming it was performing searches for documents while knowing those searches didn’t include Mrs. Clinton’s emails.

“We’re upset, and I think the courts ought to be upset, because they’re wasting our time, they’re wasting the court’s time and they’re defrauding us,” Mr. Fitton said.

In a striking move last week, the State Department agreed with Judicial Watch on a request to reopen an open-records case that involved Mrs. Clinton’s communications. Mr. Fitton said that amounted to an admission that the department realized it wasn’t performing a valid search for her records.

The House Select Committee on Benghazi is giving Mrs. Clinton more time to comply with a subpoena sent this month for all of Mrs. Clinton’s communications related to Libya.

The due date for the subpoena sent March 4 was originally March 13 but has been extended to March 27.

The committee, chaired by Rep. Trey Gowdy, South Carolina Republican, is investigating the circumstances surrounding the Sept. 11, 2012, terrorist attack on a U.S. compound in which four Americans were killed, including U.S. Ambassador J. Christopher Stevens. Mrs. Clinton, the presumed front-runner for the 2016 Democratic presidential nomination, was secretary of state at the time.

“Chairman Gowdy granted a reasonable extension because for him this is not about politics; it is about getting all relevant documents for the committee,” said committee spokesman Jamal Ware. “He still believes the best option for Secretary Clinton is to turn over her server to a neutral arbiter to independently determine what should be in the public domain. The committee has no interest in her personal emails.”

In explaining her use of a private email system and server while serving in the Obama administration, Mrs. Clinton said March 10 that she turned over more than 30,000 emails she deemed work-related. The State Department is reviewing the emails to decide what can be released to the public.

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