Before discussing "screwball dramas," is probably a good idea to discuss "screwball comedies," as the use of the word "screwball" in relation to drama can only be understood when one also understands its genesis in film lexicon in relation to comedies.
During the Great Depression, when the film industry, headquartered in Hollywood, California, sought to make films that help people forget the miseries of the time, especially the overwhelming level of unemployment, it began producing more of what were called "screwball comedies." These refer to generally light-hearted, good-natured comedies, often romantic comedies, that were both profitable for Hollywood and were highly popular with the audiences. Some examples include "It Happened One Night," "Ninotchka," and "My Man Godfrey."
"Screwball dramas" is a much more recent phenomenon, at least in terms of the use of the phrase. There is has long been a "genre" within film referred to as "dark comedies," because the story takes place against a dramatic backdrop such as war, or a depiction of a family in distress. Throughout this drama, however, there are frequent comedic moments. "The Squid and the Whale" has been considered a "screwball drama," as do the films of Spanish director Pedro Almodovar. A very recent example is "Silver Linings Playbook," which, true to the genre, depicts a serious problem -- family in distress -- but injects humor throughout the story.