After local offices in Giessen - from 1957 to 1963, he served as the Lord Mayor (Oberbürgermeister) of the city, he was elected to the parliament of Hesse in 1954.
Osswald was elected prime minister of Hesse in 1969, after the demission of Georg August Zinn.
After a financial scandal involving the public bank Helaba, Osswald resigned as prime minister of Hesse in October 1976.
A native and former mayor of Giessen, Osswald supported the creation of the city of Lahn in 1977 - born out of a merger of Gießen and Wetzlar, describing it as "Work of the Century" (Jahrhundertwerk), believing it would strengthen Central Hesse against the two major population centers of Kassel in the north of the state and Frankfurt in the south.
However, the awkwardly performed merger would prove to be unpopular and following SPD's defeat in the 1977 local elections in Lahn to CDU opposed to the city of Lahn, his successor as Minister President Holger Börner (an in-party opponent to Lahn) rescinded the merger in 1979.