Mr. King acknowledges that the organization is changing, but denies that its leadership is anti-union. He says Housing Works has maintained the spirit of community through policies that encourage workers to share their ideas and concerns.

Not all of his employees agree. Ms. Rodriguez, the staff lawyer, said that landing a job there was a dream come true. But soon after she started, Ms. Rodriguez said, she noticed things that she believed contradicted the organization’s progressive values. The tipping point for Ms. Rodriguez came about a year ago, after a colleague who was fired had to pack up her things in front of everyone in the office before being escorted from the building.

“It was really, really hard for us to believe that Housing Works would treat someone who was part of our staff and part of our family in such a heartless and sort of corporate way,” she said.

Brian Grady, a housing coordinator, said that as a gay man, he was first attracted to Housing Works because of the legacy it earned during the AIDS pandemic. Although he enjoyed his work, Mr. Grady said the workload had begun to wear him down. For months, he was the only housing coordinator for a program with nearly 3,000 clients.

“It’s difficult work,” he said. “We work with people who are going through some of the worst times in their lives and are going through a lot of really unstable situations. As workers we take a lot of that home, especially when we’re hearing these stories day after day with limited resources to help.”

Part of the problem is high turnover. Workers said that the demanding nature of their jobs meant that people were frequently coming and going. According to Housing Works’s 2019-20 strategic objective plan, its annual turnover rate is about 30 percent.

Ms. Rodriguez said the organization’s high turnover rate hurt Housing Works’s clients. She spoke about one client who was evicted and had to enter the shelter system because Housing Works could not find her a new home in time.