In 1959 he was a guide to a first winter crossing through the whole ridge of Tatra mountains which took 19 days. In 1971 he led the Polish expedition to make the first ascent to the Khunyang Chhish (7852 m) in the Karakoram. With Tadeusz Piotrowski he made the first winter ascent of Noshaq (7492 m) in the Hindu Kush (1973). It was the world's first winter climb above 7,000 m. In 1977 Zawada made the first ascent of the North Face of Koh-i Mandaras (6628 m).
In 1980 Krzysztof Wielicki and Leszek Cichy made the first winter ascent of Mount Everest (8848 m), the highest mountain on Earth. Zawada led the expedition. It was the first ascent in winter of a peak above 8,000 m.