The Governor and legislature should be fired, voted out of office, removed from their duties and competent people elected, appointed or otherwise brought in as replacements.
Since the time of ancient Roman politicians and governments have found ways to balance budgets even honorably on occasion. A student of history may recall, perhaps from reading Livy, that the Roman Senate in times of public distress appointed a Dictator for a year at a time to address a present public danger. That is an illustration intended to express the point that other politicians have faced similar and greater challenges and found ways to respond competently. Alaska politicians are not experiencing anything new or unprecedented.
Romans also appointed a master of the horse, yet the military situation is more of a problem for the federal government than the state regarding the budget crisis. The Alaska legislature may not need to find a benevolent dictator to solve the state budget problems, yet the comparison is useful to underscore how ineffective they are at getting the job done.
It would be possible to take extreme economic measures to directly balance the budget if it were really needed. One could just cut the state budget 50% and triage the surviving financial needs. Like poor people that can't afford a home the legislators could meet outside and live in a tent city on some existing state land saving oodles of money. If it is really needed to be cheap they could spend as little as Alaskans in government before the pipeline.
I slept outside on the ground a winter in Anchorage myself and know its quite possible to survive and get written work done. Politicians and constituents could share a coffee over a propane stone and have a sense of shared sacrifice. If they weren't overly comfortable bureaucrats could balance the budget and leave the PFD alone.
If the state of Alaska were hit by a real series of disasters rather than an economic one of their own making they could cut the budget 50% and/or redirect the money to the necessary directions that would address right issues.
For bureaucrats worried about paper, front loaders could stop at various government agencies and collect the stuff in garbage bags and move it into dump trucks. Then it could be removed and transported to some cold storage facility with a vacuum to prevent degradation of material. Everything should already be digitally recorded though, and the paper redundant.