RIVIERA BEACH, Fla. — Florida’s Secretary of State ordered machine recounts in three statewide races Saturday, as tallies submitted by the state’s 67 counties showed the contests for Senate, governor and agriculture commissioner were too close to call.

Recounts were also ordered in a state Senate race and two contests for the state House.

“Florida has never had a full statewide recount. It’s about to have three,” Andrew Weinstein, the national co-chair for the Democratic Lawyers Council, said on Twitter. “Buckle up.”

Some candidates who saw comfortable margins diminish since Tuesday, as heavily Democratic southern counties continued to process mailed and problem ballots, cried fraud and filed lawsuits.

Gov. Rick Scott, whose margin in the race for the Senate narrowed to less than 13,000 votes, denounced the embattled Broward County elections supervisor, Brenda Snipes, who, the campaign said Friday night, still refused to confirm whether she had counted all the ballots. Dr. Snipes was forced to admit that she had inadvertently tabulated about a dozen rejected ballots, which only fueled Republican accusations that her office had botched the vote-counting process.

“Three days after Election Day, the vote tally continues to change and Supervisor Snipes still refuses to explain where and how the new votes came to light,” the campaign said in a statement. “The public deserves a clear and direct answer.”

He urged sheriff deputies to be on alert for any reports of vote-rigging.

As of noon on Saturday, the deadline for the state’s counties to hand in unofficial results, three statewide races remained under the 0.5 percentage point margin for a legally required machine recount: the Senate race between Mr. Scott, a Republican, and Senator Bill Nelson, a Democrat; the governor’s race between Ron DeSantis, a Republican, and Andrew Gillum, a Democrat, and the commissioner of agriculture race between Nikki Fried, a Democrat, and Matt Caldwell, a Republican.

Secretary of State Ken Detzner, formally ordered the recounts on Saturday afternoon. The new tallies were expected to begin as early as Saturday afternoon in the state’s largest counties, Miami-Dade, Broward and Palm Beach. Other counties could also proceed immediately, though many were expected to wait until Sunday to begin.

Florida wasn’t the only state still trying to determine the outcome of Tuesday’s elections. In the Georgia governor’s race, Brian Kemp, the Republican, was ahead of his Democratic opponent, Stacey Abrams, by about 63,000 votes.

In Arizona, Kyrsten Sinema, the Democratic candidate in the state’s uncalled Senate race, expanded her lead on Friday to 20,203 votes over Martha McSally, the Republican contender.

Each county in Florida will have until Thursday to run all of its ballots through counting machines again. At that point, any race that remains within a margin of 0.25 of a percentage point or less will have another three days, until Nov. 18, to conduct a manual recount.

Manual recounts seem almost certain in the races for Senate and commissioner of agriculture, which are already within that quarter-point margin.

A manual recount does not mean every ballot is counted by hand.

Only the votes that come up as an “undervote” or “overvote” get pulled for manual review. For example, if a voter had put a check mark next to a candidate’s name instead of filling the circle out completely, the machine could have missed it.

See How the Close the Results Are in Florida, Georgia and Arizona

The races for Senate and governor may head to a recount or a runoff.

In cases where the machine detects that a person actually chose two people in the same race, a team of election workers looks at the ballot to see if the voter’s intention was clear. The person could have crossed out one candidate’s name, so that ballot would likely be counted.

But several issues could arise during the process. Older counting machines might be unable to conduct an unprecedented three statewide recounts simultaneously, making it impossible to meet the state’s deadline. If a county is unable to complete a recount in any particular race, Saturday’s unofficial results from that county would stand for that race.

Brian Y. Silber, a lawyer, went through an exhaustive manual recount this summer when a Broward County judge he represented seemed to have lost on Election Day — only to wake up the next morning to find that he had taken the lead. It took two days to get a new result, he said, and that was with about a quarter of the number of ballots that now must be reviewed.

“I would be shocked, really shocked, if there was evidence of fraud, conspiracy, anything illegal or evil,” Mr. Silber said. “What I genuinely believe is that elections officials, for whatever reason, are not getting it done on time. That’s a combination of poor management, underfunding and understaffing.”

The problem has been going on for years, he said.

He said Republicans had a savvy — albeit misleading — strategy to convince supporters in advance that any loss would be attributable to misdeeds by Democrats.

“It’s really smart on the G.O.P.’s part,” Mr. Silber said. “They know there is no evidence of fraud.”

Daniel A. Smith, chairman of the political science department at the University of Florida, said 41,000 Floridians requested mail-in ballots from overseas, so elections supervisors were simply inundated with ballots to count after Election Day.

“My position quite honestly is there is very low likelihood of fraud going on anywhere,” Mr. Smith said. “There is a capacity issue when so many ballots come in on Election Day.”

Mr. Scott claimed fraud in Broward County even though the state was monitoring Dr. Snipes’s office during the election. The state’s division of elections assigned two staffers to watch how the election was administered, visit polling places and observe the preparation of voting equipment and procedures.

The monitors made no reports of fraud.

“Our staff has seen no evidence of criminal activity at this time,” Sarah Revell, a spokeswoman for the division of elections, said on Saturday.

The odds that Mr. Gillum, who trails by about 33,600 votes, or 0.41 percentage points, and even Mr. Nelson, who is behind by less than 13,000 votes, or about 0.15 percentage points, will find themselves on top after a recount seem low, according to veterans of Florida’s presidential recount in 2000.

Marc Elias, Mr. Nelson’s attorney, has maintained that a machine error might still account for fewer votes for Mr. Nelson — an issue that would only be caught in a manual recount.

“Ultimately, the ballots are what they are,” he said. “The votes are what they are.”

Frances Robles reported from Riviera Beach and Patricia Mazzei reported from Miami.

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