DOVER — The Joint Finance Committee received a look at Gov. John Carney’s budget proposal Monday, examining education funding, health care costs and the possibility of shifting expenses to local governments.

The governor’s plan, announced in March, used an even mix of cuts and taxes to bridge a $386 million projected shortfall. That gap has since grown by $8.9 million after new financial estimates were released last week, although budget officials have cut $13.7 million in expenses.

Monday included a brief discussion on the structure of government spending in the state, with one senator advocating for lawmakers to seriously consider how they want to shift costs onto the counties, municipalities and school districts.

Gov. Carney’s proposal would move 5 percent of the cost of school transportation, which the state currently covers 90 percent of, to the districts. It would make other cuts to education funding as well and allow districts to raise property taxes without referendums.

“We’re making a decision that we think homeowners in Delaware should pay higher property taxes in order to make it possible for the state government to expand its spending and expand its programs,” Sen. Brian Bushweller, D-Dover, said.

Decision-makers need to consider whether they want to move in that direction and, if so, what large-scale changes should be made, he suggested.

Members took a look at health care costs, which are growing much faster than revenue, as well Monday. State employees could soon be forced to pay more for their care, and while the form of those costs — higher premiums or deductibles, for instance — has not been decided, Gov. Carney’s budget called for moving $6.5 million to state workers. His idea also would save $3.5 million by ending a provision that allows two married state employees to receive health care for $25 a month.

The state covers about 90 percent of employee health care costs.

While a similar proposal made two years ago was unpopular and ultimately did not go through, budget officials have stressed the high costs of health care need to be curbed.

“Whatever savings we don’t make up here, we have to make up somewhere else,” Office of Management and Budget Director Mike Jackson said.

Education growth is posing problems for budget writers as well, with the state being forced to hire more teachers and devote more resources to education, largely due to a rapid increase in the number of special education students.

More children are being diagnosed with learning differences, and others are transferring in from private schools or even another state.

The governor’s budget does include money for teacher step increases, which are regular pay raises educators receive. The increases, Mr. Jackson said, help attract and keep teachers.

Some other state employees, such as nurses and capitol police officers, also receive steps. However, Mr. Jackson differentiated between those and the kind given to teachers. Non-teacher step increases have been suspended in the past and could be frozen again this year.

The state is currently in the process of negotiating with correctional officers, who are seeking higher pay and other changes. A longstanding issue for officers, pay and staffing levels have boiled over since an inmate uprising at the James T. Vaughn Correctional Center in February. The incident left one officer dead.

Mr. Jackson said the state began contract discussions with officers last week. However, changes would not take place until July 1, 2018, unless a deal is reached in the next two months, allowing the Joint Finance Committee to enact the adjustments through the budget bill.