You would need to legally reside in the Netherlands and demonstrate that you are sufficiently integrated. At that point you could naturalize and retain your existing citizenship (provided that South African law does not strip you of citizenship if you naturalize elsewhere).
The general requirements are listed at the below link:
Normally, you must have resided in the Netherlands for either 3 or 5 years prior to naturalization, but according to the below link, you are exempt from this requirement as you and your husband have already lived together for 3 years:
This page explains how, as the spouse of a Dutch citizen, you are not obliged to give up your existing citizenship when naturalizing:
As your husband is a Dutch citizen, you can obtain a residence permit on that basis.
The biggest obstacle may be demonstrating you are integrated, which involves demonstrating proficiency in Dutch and some familiarity with Dutch cultural norms and values. If you can speak Afrikaans, then picking up Dutch is typically not that hard so the norms and values may be where you must spent your effort. You can find some details on the exams that prove you are integrated here: