I have a business class award ticket on Delta/South African Airways for travel from DFW- Cape Town.
I want to be on the nonstop flight from ATL to Cape Town. In order to get the nonstop I'd have to leave a day earlier which is no problem for me. In addition, I'd like to spend an extra day in South Africa in order to get the most out of it. Delta tells me if I leave a day earlier there are no award seats anywhere on the early morning flight from DFW-ATL, but the ATL Cape Town flight is available.

If I leave a day earlier to spend an extra day in SA and get on the nonstop I have the option of paying $199(3 day advance purchase) for a one way DFW-ATL ticket and then paying $100 penalty for changing my award ticket. OR, I can just wait and hope an award seat from DFW-ATL opens up. Of course if I wait for a DFW-ATL award seat to become available the ATL-Cape Town seat may have been taken.

What should I do?

Here's my uninformed opinion. If you want the extra time, the amount you are discussing is probably small in the scheme of things. And to quote the received wisdom of the ages: A bird in the hand is worth two in the bush.

How many times are you going to get to South Africa? Might as well make this trip long enough to see what you want.