AUGUSTA, Me. — Government workers marched outside the State House here chanting, “Do your job!” on Monday as Maine kept children’s caseworkers at home and shut down other offices deemed nonessential.

A standoff over a tax increase left Illinois teetering on the edge of a potentially devastating credit downgrade. And a deadlock over a raid on the funds of New Jersey’s largest health insurer kept the state’s parks and beaches closed for a third straight day, though lawmakers were newly hopeful of a settlement late Monday.

Stalled negotiations have left at least eight states without budgets several days into a new fiscal year. But the effects have been felt most starkly in three of them, where outspoken Republican governors are clashing with Democratic-controlled legislative chambers.

If those three impasses highlighted the perils of divided state government, which has grown increasingly rare in an era of single-party rule, they also underscored the limits of the Republican Party’s ascendancy on the state level. The governors share more than a party label: They are, to a man, proud provocateurs who have tried to enact conservative policies in blue states through badgering and political intimidation.

Govs. Chris Christie of New Jersey and Paul R. LePage of Maine have routinely tried to humiliate and overwhelm their adversaries through force of personality, often projecting bluster through the news media, while Gov. Bruce Rauner of Illinois has used his vast personal fortune to run television ads attacking legislators who refuse to support his agenda.

State Senator Richard J. Codey of New Jersey, a Democrat who served as acting governor from 2004 to 2006, said that those sorts of imperial attitudes were bound to end in ruin. The shutdowns, he said, reflected a collapse in the traditional give-and-take between governors and legislative leaders, who have often managed to cooperate across party lines.

“As long as there’s reasonable people as the heads of the legislature and the governor, you’ll get it done,” Mr. Codey said. “There’s no excuse for this.”

In New Jersey, Mr. Christie’s clash with legislators stems from his demand for more authority over the state’s largest health insurer, Horizon Blue Cross Blue Shield. “He can’t get his way all the time,” Mr. Codey fumed. For his part, Mr. Christie has blamed the Democratic speaker of the State Assembly, Vincent Prieto, calling him “the one who dug his heels in.”

A majority of states — 33 of them — are now controlled by a single party, and some of those also hit budget impasses this year. But provisions in many of those places, such as extending funding at current levels or setting temporary budgets until deals can be made, have spared residents, for the most part, from the effects.

In Wisconsin, where Republicans control the governor’s office and the State Legislature, disagreements over transportation funding have slowed lawmakers. In Rhode Island, which is entirely under Democratic control, the House has yet to vote on a $9.2 billion budget plan passed by the Senate on Friday. And in Oregon, Gov. Kate Brown, a Democrat, and the Democratic-leaning Legislature have sparred over how the state taxes businesses, but lawmakers appeared to be headed toward an agreement this week.

In states like Maine, however, the effects were immediate. Though state parks and emergency services were kept open, motor vehicle offices were closed, as was the lottery office. And the operations of agencies like the Office of Child and Family Services and the Department of Labor were curtailed, as thousands of workers were idled and lawmakers inched toward a deal.

“Days like today, we regularly have caseworkers that would go out to check on the children,” said Dean Staffieri, the vice president of a state employees’ union and a supervisor in a child protective services office, which he said had only a small crew on call for emergencies on Monday.

For residents, the state’s first shutdown in decades was jarring. Danielle Sirois, a cashier at a Christmas Tree Shops store in Augusta, said she worried for the families of state employees. “I think they should all come together,” Ms. Sirois said of Maine’s lawmakers. “The world is scary to me right now because everybody is so at odds.”

Maine’s combative governor, Mr. LePage, said he would allow state employees to take a day of paid leave on Monday, but many of the workers gathered at the State House wondered how he could do that when there was no budget in place.

At the heart of the fight, at the moment, is Mr. LePage’s vow not to sign a budget this year with any tax increases.

Late last week, Senate Republicans and the Democrats who control the House agreed on a compromise budget. In a major concession by Democrats, it would repeal a voter-approved tax surcharge on the state’s highest earners, which had been expected to generate more than $300 million for education. But the bill also contained a roughly $20 million increase in the state’s lodging tax — as had the budget Mr. LePage first proposed.

On Monday, as lawmakers considered a new compromise — which still contained a lodging tax increase — Mr. LePage used a Facebook video to express his displeasure.

“If they vote a budget out today without a tax increase, this shutdown is over,” he said. “But if they do not and they put a tax increase, I cannot put my fingerprints on this bill.”

State Senator Roger Katz, a Republican who backed the budget deal, said the shutdown was the embarrassing result of negotiations with a moving target. The same lawmakers now incensed by the lodging tax, he said, had different complaints just days ago.

“The tradition of consensus budgeting has really begun to fall apart here, as it has nationally,” Mr. Katz said. “I think for some, it’s about ego, it’s about wanting to be relevant, and it’s about wanting to have one’s fingerprints on the result. For some people, the fight is more important than the win.”

In Illinois, the Democratic-controlled House approved a tax increase on Sunday, moving tantalizingly close to the end of a more than two-year budget standoff that has sullied the state’s reputation, caused $15 billion in unpaid bills and denied services to thousands of poor and elderly residents.

Fifteen Republicans — most of them from downstate Illinois — joined 57 Democrats to vote for the measure, which would raise the personal income tax rate to 4.95 percent from 3.75 percent, and the corporate income tax rate to 7 percent from 5.25 percent. The increases would generate about $5 billion.

Some Republicans sounded anguished over their vote, and how it might affect their political careers. “For me, right here today, right here, right now, this is the sword I’m willing to die on,” Representative Michael Unes of East Peoria said. “And if it costs me my seat, so be it.”

But Governor Rauner, a Republican who had never held elected office before he won his seat in 2014, has already declared his opposition and said he would veto the measure if it passed the Senate and reached his desk. Mr. Rauner has pushed for changes like a freeze on property taxes and cuts to the cost of workers’ compensation.

“Illinois families don’t deserve to have more of the hard-earned money taken from them when the Legislature has done little to restore confidence in government or grow jobs,” he said in a statement. “Illinois families deserve more jobs, property tax relief and term limits.”

Sunday’s vote revealed a sense of weariness in Springfield over a budget impasse that entered its third year on Saturday, a first for any state. The situation became especially dire in the last few weeks, as the start of the new fiscal year approached. Credit ratings agencies threatened to downgrade Illinois’s rating to junk status, an unprecedented move that would make borrowing even more costly.

Don Rose, a longtime political consultant in Chicago, said he was stunned by the inability to find common ground in a state where, over the years, Republican governors have often worked out deals with Democrats in the legislative chambers.

“I just do not remember anything resembling this, and I’ve been around a while,” Mr. Rose said. “When control was shared, we’ve always gotten along. It was never bitterly divided like this.”