In April 2019, David Ginsberg, a Meta executive, emailed his boss, Mark Zuckerberg, with a proposal to research and reduce loneliness and compulsive use on Instagram and Facebook.

In the email, Mr. Ginsberg noted that the company faced scrutiny for its products’ impacts “especially around areas of problematic use/addiction and teens.” He asked Mr. Zuckerberg for 24 engineers, researchers and other staff, saying Instagram had a “deficit” on such issues.

A week later, Susan Li, now the company’s chief financial officer, informed Mr. Ginsberg that the project was “not funded” because of staffing constraints. Adam Mosseri, Instagram’s head, ultimately declined to finance the project, too.

The email exchanges are just one slice of evidence cited among more than a dozen lawsuits filed since last year by the attorneys general of 45 states and the District of Columbia. The states accuse Meta of unfairly ensnaring teenagers and children on Instagram and Facebook while deceiving the public about the hazards. Using a coordinated legal approach reminiscent of the government’s pursuit of Big Tobacco in the 1990s, the attorneys general seek to compel Meta to bolster protections for minors.

A New York Times analysis of the states’ court filings — including roughly 1,400 pages of company documents and correspondence filed as evidence by the State of Tennessee — show how Mr. Zuckerberg and other Meta leaders repeatedly promoted the safety of the company’s platforms, playing down risks to young people, even as they rejected employee pleas to bolster youth guardrails and hire additional staff.