“You can’t just look at governors these days,” Mr. Norquist said. “You’ve got to look at the legislatures. The legislature in North Carolina is much more pro-growth and anti-tax than the governor.”

The financial picture for states has stabilized, with five straight years of revenue and spending increases. But when those numbers are adjusted for inflation and population growth, they are still far below their pre-recession peak. The National Association of State Budget Officers recently warned that revenue growth was not strong enough to both maintain basic services and meet the growing demand to spend more on such things as higher education and prisons.

And the recovery is uneven: In 20 states, revenues in the current fiscal year are lower than expected, according to a December survey by the association.

“As new governors and legislators begin the session, they’re realizing that money is tight — there’s not enough to go around even if you just want to do the basics, like increase K-through-12 spending and fully fund Medicaid,” said Scott D. Pattison, the association’s executive director. “I think what’s going to happen is that elected officials want to implement tax cuts, but they can’t do anything significant or dramatic because the growth is so limited.”

The governors, including ones returning to office and some new ones, are responding in different ways to this unsteady ground.

In South Dakota, Gov. Dennis Daugaard proposed an increase to the gas tax and several taxes and fees related to motor vehicles because highway funding is falling short. Gov. Gary R. Herbert of Utah has increased the tobacco tax, proposed extending it to e-cigarettes and said he was open to an increase in the gasoline tax.

Gov. Nikki R. Haley of South Carolina, a Republican, said she would allow the state to raise its gas tax, a move she had previously promised to veto, but only if the legislature cut the income tax by almost a third and reformed the State Transportation Department. And Gov. Doug Ducey of Arizona, a Republican elected last November, has moved away from his campaign pledge to eliminate the income tax, which provides about one-third of the state’s budget.