Oswald was born and raised in the St. Vital neighbourhood of Winnipeg, Manitoba, and was a teacher and school administrator for fifteen years before entering politics.
She initially taught English, later serving as vice-principal at Victor Mager School, Winnipeg in the Louis Riel Division.
Oswald was part of the NDP's historic breakthrough in south-end Winnipeg in the provincial election of 2003, defeating incumbent Progressive Conservative Louise Dacquay in the riding of Seine River, 4,314 votes to 3,582.
In August 2004, Oswald was appointed to lead a task-force committee looking for ways to encourage physical activity in the province's youth.
Oswald resigned her cabinet position on November 3, 2014, along with Jennifer Howard, Erin Selby, Stan Struthers, and Andrew Swan due to concerns about Premier Selinger's leadership.