WASHINGTON — At her annual budget address this month, Gov. Kristi Noem, Republican of South Dakota, blamed President Biden’s economic policies for rising prices, derided the “giant handout” of federal stimulus funds and suggested that she had considered refusing the money over ideological objections.

But like many Republican officials, Ms. Noem has found it hard to say no to her state’s share of the $1.9 trillion pandemic relief aid that Democrats passed along party lines in March.

Ms. Noem explained to fellow legislators how critical those federal funds were to South Dakota and outlined how she would use some of the nearly $1 billion slated for her state to invest in local water projects, make housing more affordable and build new day care centers. For those questioning her choice to take the money, Ms. Noem, who has opposed Covid restrictions including shutdowns and mask mandates, said any pandemic-relief funds she rejected would have just gone to other states.

“It would be spent somewhere other than South Dakota,” Ms. Noem said. “The debt would still be incurred by the country, and our people would still suffer the consequences of that spending.” No state has declined the relief money, and if they had it would go back to the Treasury Department, not to other states.

Republican leaders across the country have been engaged in a similarly awkward dance over the past few months as they accept — and often champion — money from the $350 billion bucket of state and local aid included in the stimulus bill, which passed Congress without a single Republican vote. In some states, like Ohio and Arizona, Republican governors are spending the funds while attempting to undercut the law that allowed the money to flow. Other governors are faulting Congress for not giving their state enough money.

And, like their counterparts in Congress, many Republicans have blasted Mr. Biden’s stimulus bill for fueling inflation, even as they take the funds, and criticized Democrats for pushing for additional government spending plans.

“I urge President Biden and Democrats in D.C. to turn off the spigot of out-of-control spending and get inflation under control,” said Gov. Greg Gianforte, Republican of Montana, whose state has used some of its $906 million in stimulus money to invest in nursing homes and return-to-work bonuses.

Gov. Ron DeSantis, Republican of Florida, complained last week that the federal formula for allocating money to states based on their jobless rate had essentially penalized Florida for not imposing lockdowns and allowing businesses to remain open during the pandemic.

“I think you’d have to acknowledge that we got the short end of the stick compared to these other states,” Mr. DeSantis said.

Florida, which was allotted a total of $8.8 billion, has so far received about $3.4 billion, which Mr. DeSantis said would go toward infrastructure, transportation and work force retention. The governor justified keeping the money by arguing that the federal government fueled economic disruption with shutdowns and vaccine and mask mandates that he opposed.

Despite his complaints, the cash cushion could help Florida build as much as $17 billion in reserves by the end of next year, according to Mr. DeSantis, and allow the state to afford to pay for priorities that are unrelated to the pandemic. Mr. DeSantis proposed a gas tax holiday and an $8 million program to remove “unauthorized aliens” from of the state. The money for that program would come from the interest generated by the state and local recovery funds, a spokeswoman for Mr. DeSantis said.

A Treasury spokeswoman said that the agency did not preapprove uses of funds but that any funds used “in violation of eligible uses” of Treasury’s rules could be clawed back by the federal government.

States, which have until 2026 to spend the stimulus money, are getting their share of federal funds at a moment when budgets are recovering faster than expected, with many governments awash in cash and announcing large surpluses.

A November expenditure report from the National Association of State Budget Officers found that state revenues are up 12.8 percent this year after declining in 2020 for the first time in a decade. State spending of federal funds has surged by 35.7 percent this year, and more money is on the way, as many states will get their second tranche of relief money in 2022. The recent passage of the $1 trillion bipartisan infrastructure law will also send more federal funds to local governments.

It is a far different picture than what state officials expected at the beginning of the pandemic, when budget officers across the United States warned of dire shortfalls as businesses shuttered, workers lost their jobs and health costs soared. But a stronger than expected economic recovery and trillions of dollars of relief money have left states with a new problem: how to spend it.

State officials say the biggest challenge has been figuring out how to get the federal money out the door amid a complex set of spending rules that has yet to be finalized by the Treasury Department.

“It’s like the python that ate the rat,” Brad Whitehead, a nonresident senior fellow at the Brookings Metropolitan Policy Program, said of the struggle to shepherd so much federal money into state projects. “You need all those calories, but it’s hard to digest it all at once.”

The Treasury Department gave states broad discretion over how the stimulus money can be deployed, but imposed limits on using funds to shore up public pension programs and restricted states from using relief funds to subsidize tax cuts. The tax cut prohibition angered several Republican governors, who argued it infringed on state sovereignty, and has led to a thicket of lawsuits.

Among those challenging the restriction is Ohio, which was awarded nearly $5.4 billion of state aid through the American Rescue Plan. Gov. Mike DeWine, a Republican, opposed the entire package and, after it passed, his state took a leading role in litigation contending it was unlawful to put conditions on the relief money that prohibited states from using it to finance tax cuts.

The lawsuit is still making its way through the courts, but by June, Mr. DeWine signed legislation to use more than $2 billion of the federal funds to replenish the state’s jobless benefits fund, to improve water and sewer quality and to improve pediatric behavioral health facilities.

Texas announced in October a raft of plans to start spending some of its nearly $16 billion in federal aid, unveiling major investments in broadband, rural hospitals and food banks. Yet the state, which received the second largest allotment of funds in the country, also said it was hoping to use some of the funds to slash property taxes and that, despite the prohibition against doing so, it was setting aside $3 billion for “future tax relief.”

The most contentious use of federal funds this year has been in Arizona, where Republican Gov. Doug Ducey used relief money to roll out two education programs intended to undercut mask mandates that were imposed by some school districts. A $163 million program provides up to $1,800 in additional funding per pupil in public and charter schools that are “following all state laws” and open for in-person instruction. Schools that required masks would not be eligible.

A separate $10 million program provides vouchers worth up to $7,000 to help poor families leave districts that require face coverings or impose other Covid-related “constraints.”

The Treasury Department warned Mr. Ducey in October that the state could lose some of its $4.2 billion if it did not change the policy. Arizona rebuffed the request and a senior Treasury official said that an administrative process is now underway to claw back some of the funds.

A spokesman for Mr. Ducey did not respond to a request for comment.

White House officials said that despite some disagreements about the relief money, governors and their staffs have privately been working well with the Biden administration.

“I’ve had direct conversations with virtually all of the Republican governors or their top officials, and to the one, they have been constructive, nonpolitical, nuts and bolts conversations about how they can best use their American Rescue Plan funds for things like broadband, schools, water and work force development in a way that meets the needs of their state,” said Gene Sperling, who is Mr. Biden’s pandemic relief czar.

Other aspects of the relief package remain in limbo since it was enacted, including a $4 billion debt relief program aimed at helping minority farmers. That remains mired in litigation brought by some white farmers and conservative groups such as America First Legal, which is run by the former Trump administration official Stephen Miller. They argue that the program, which was a centerpiece to the Biden administration’s racial equity agenda, unfairly excludes white farmers because of their race.

The debts of minority farmers, who faced years of discrimination by the Agriculture Department, have yet to be forgiven.

John W. Boyd Jr., the president of the nonprofit National Black Farmers Association, said he found it wrong that states like Texas, where the agriculture commissioner is suing to block the debt relief, are grappling with how to spend their relief money while Black farmers cannot access their aid funds and are facing foreclosure.

“I think it’s terribly unfair,” Mr. Boyd said. “But it’s a continuation of what we’ve endured in this country.