Compatibility summary
You will need a travel adapter in Zambia; plug types used there differ from South Africa.
Plug comparison
South Africa plug types: C D M N
Zambia plug types: G
There is no common plug type between South Africa (C D M N) and Zambia (G). You need a travel adapter.
Voltage comparison
South Africa: 230V, 50Hz
Zambia: 230V, 50Hz
Voltage difference is within a range many devices tolerate; check your appliance label.