Zawada started his career at his local club, DKS Dobre Miasto. Later he moved to
OKS 1945 Olsztyn. In 2012, he joined German club
VfL Wolfsburg, aged 16. Two years later he made his senior debut for the reserve team. In January 2016, he joined Dutch club
FC Twente on a loan. On 26 January 2017, he was identified as surplus to requirements at VfL Wolfsburg and was sold to
2. Bundesliga side
Karlsruher SC for an undisclosed fee. In December 2017, it was announced that Zawada would join Polish
Ekstraklasa club
Wisła Płock for the second half of the 2017–18 season. In October 2020, he joined the Polish Ekstraklasa club
Raków Częstochowa. In March 2022, following a short-lived stint at Korean side
Jeju United, he joined Polish club
Stal Mielec until the end of June. In August 2022, he joined
A-League side
Wellington Phoenix (a New Zealand-based side) on a two-year contract. Zawada enjoyed his most successful season to date during the
2022–23 campaign, scoring a career-high 15 goals for the club. He followed it up with 7 goals in 14 league games in the
2023–24 season, during which Zawada saw limited playing time due to injuries. On 4 July 2024, Zawada returned to Dutch football by joining
Eredivisie club
RKC Waalwijk on a two-year deal. On 11 July 2025, Zawada signed a three-year contract with another Eredivisie side
FC Groningen. ==Career statistics==