Allison Marotta, a 25-year-old New Yorker with Type 1 diabetes, said that in a few weeks she will no longer be carried under her parents’ insurance. She isn’t eligible for Medicaid and said she can’t afford a state-sponsored program. Her monthly care costs will exceed $2,000 a month without insurance for insulin alone, with supplies and doctor’s appointments quadrupling that number, she said.

“I will not be able to afford this,” Ms. Marotta said through tears. “After my 26th birthday I will no longer have access to my team of medical doctors and I will begin to ration insulin and medical supplies.”

Advocates hope that the onslaught of personal and painful stories shared during the hearings—three have happened so far and at least one more is expected—will build momentum to pass the legislation. They say the plan would provide New Yorkers with complete health coverage without deductibles, copays or out-of-network charges.

Versions of the Health Act have previously been taken up and passed by the Assembly, which has long been dominated by Democrats, but there was no vote on the bill during this year’s session—the first in a decade during which Democrats also controlled the state Senate.

During a press conference before the Bronx hearing, state Sen. Gustavo Rivera, chair of the senate’s health committee, said this year will be different. The Health Act, he said, will be a “central part” of the conversation of the upcoming legislative session, which begins in January.

“We know that health care is something that we all struggle with,” he said. “And now we are saying that we have a solution to deal with that in the state of New York.”

Still, there are political roadblocks at the Capitol in Albany. Senate Majority Leader Andrea Stewart-Cousins, a Democrat from Yonkers, hasn’t embraced the bill, as she attempts to balance the needs of moderate Democratic senators from suburban areas—whose electoral victories helped propel the party into the majority—with those of more liberal members who support the bill.

In a statement, Ms. Stewart-Cousins committed to discussing the bill, and said “Improving our broken health-care system is a Senate Democratic priority and a major concern.” A spokesman for Assembly Speaker Carl Heastie, a Democrat from the Bronx, said he would discuss the issue with his members.

State Sen. Patrick Gallivan, a Republican and ranking member of the Senate’s Health Committee, has called the Health Act highly impractical. “No taxpayer-funded program can afford unlimited and unrestricted health care,” Mr. Gallivan said earlier this month.

Gov. Andrew Cuomo has said he wants to expand access to health insurance, but has thrown cold water on the Health Act. The governor has questioned whether a single-payer system could exist on a state scale as opposed to a national one, and said he doesn’t believe the federal government would give the necessary Medicaid waivers to implement such a bill.

Leanne Politi, a spokeswoman for the Realities of Single Payer, a coalition of health insurance and business groups that supports universal coverage but not a single-payer system, said the testimony and questions raised at the public hearings has only reinforced members’ opposition to a statewide, government-run single-payer system. Many people currently uninsured are eligible for low or no-cost health coverage right now.

“State lawmakers should explore other ways to get the remaining individuals covered as well as addressing issues which will lower the costs of care making coverage more affordable,” said Ms. Politi. “Let’s fix what is broken instead of throwing out the entire system.”

Bronx resident Helene Reed explained to lawmakers at last week’s hearing how two children she cares for lost health insurance, despite qualifying for a state program.

Andrew P. Raia, a Republican assembly member from Long Island, said the problems they are encountering are with the current Medicaid system.

“Reach out to your state legislators, let them be your advocate,” Mr. Raia said.

“And then what happens?” asked Ms. Reed. “Tell me, what can they do to help me?”

“I would reach out to your state representatives,” said Mr. Raia, “and see if they can help you cut through the red tape.”

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