New York Times Co. said its third-quarter earnings fell sharply as print advertising dropped 19%, the latest publisher to signal that spending on newspaper ads is drying up even further.

Publishers have made plans to further trim their staffs and shift resources to more digital initiatives as advertising demand is forecast to remain weak. The New York Times said its overall advertising revenue decreased 7.7% in the third quarter, and the company expects a similar decline for the fourth quarter. Digital advertising sales, meanwhile,...