WASHINGTON—Finalized rates for big health insurance plans around the country show the magnitude of the challenge facing the Obama administration as it seeks to stabilize the insurance market under the Affordable Care Act in its remaining weeks in office.

Market leaders that are continuing to sell coverage through HealthCare.gov or a state equivalent have been granted average premium increases of 30% or more in Alabama, Delaware, Hawaii, Kansas, Mississippi and Texas, according to information published by state regulators and...