Mr. Walker’s recovery plan would take more from residents through the income tax and would give them less as well, by changing the formula under which the dividend is paid. The income tax would be 6 percent of the amount an Alaskan currently pays in federal taxes, so a person who owed $10,000 to the Internal Revenue Service would also need to write a $600 check to Alaska.

Dividend payments would be tied directly to royalties that decrease or increase with oil production. Because oil production is down, next year’s payout would be cut by roughly half under the proposal, to about $1,000 a person. The governor would also raise taxes on alcohol and tobacco and would collect new taxes from the fishing, mining, energy and tourism industries.

With so much pain to spread around, Mr. Walker, a former Republican, will clearly have a fight on his hands in the Republican-controlled Legislature. He was elected in 2014 on a “unity ticket” with a Democrat as his lieutenant governor, defeating the Republican incumbent, Sean Parnell — a perceived betrayal for which many Republicans have not forgiven Mr. Walker.

But he also has some powerful allies.

“We’ve had it awfully good for a long time, and if we want to protect that, we’re going to have to make some hard choices,” said Ronald Duncan, the president and chief executive of GCI, a telecommunications company that is one of the state’s largest non-oil businesses.

Mr. Duncan is organizing a group to push for hard choices with a statewide publicity campaign that will start next month. The campaign, aimed at residents as well as the Legislature, is still being shaped, Mr. Duncan said. But his enthusiasm for the governor’s message in recent weeks — that residents will have to be less reliant on oil companies and pick up more of the burden themselves, as they did in the past — was clear. Mr. Duncan said he thinks a broader economic recession is inevitable next year if Alaska’s budget is not stabilized.

“I’m a fan of what the governor has done here,” he said. “The fact that he has relatively good approval ratings in the general population is helpful in that regard. The place in the state where the governor has absolutely horrendous approval ratings is in the Legislature.”