First, let me say that you have omitted two critical facts: Does your future husband reside in Germany? Do you plan to settle with him there?
If the answer to both questions is yes, then your presence in Germany would be subject to German national immigration law, which is somewhat more restrictive than the EU's free movement provisions. You would still benefit from the EU's free movement law when you are in EU and Schengen1 countries other than Germany.
Do you have to leave Schengen zone to validate a new Schengen visa after marrying to an EU citizen?
In general, no. If you're planning to settle in Germany, however, you would need to comply with German law, and I do not know precisely what it has to say about this. There are several countries whose citizens may in general apply for residence permits in Germany without first getting a visa. I do not know for certain whether this applies to family visas, but if it does then you should not need to leave the Schengen area. A citizen of Mexico, on the other hand, probably would need to do that. (The countries are the USA, Australia, Canada, Israel, Japan, New Zealand, Switzerland, and South Korea; see Residence Visa / Long Stay Visa on the German government site.)
Once we get married, is it better to leave the Schengen zone and re-enter and present our marriage certificate?
I'll be 'joining' him as a spouse of an EU citizen. I believe this will make us be free from tied to the 90/180 Schengen rule.
That's true, unless your husband resides in Germany. In that case you will be free from the 90/180 rule in every Schengen country other than Germany. In Germany, your presence would be governed by German domestic law, so you would need a residence permit to exceed the 90/180 rule.
In other visa types, I know if you change the status of your visa (ie. Student visa to work visa) you have to leave the country and re-enter the country with a fresh stamp.
That's not actually true. Different Schengen countries have different views of that, but there's nothing in the Schengen regulations that requires it. Regardless, once you're married, the Schengen regulations will no longer apply to you for the purpose of regulating your stay when you're traveling with your husband in the Schengen area (except, if your husband resides in Germany, when you're in Germany).
Instead, you will enjoy a derivative right to accompany him as he exercises freedom of movement anywhere in the EU or the Schengen area (except for Germany if he resides there). That right flows from the EU treaties, not from any document such as a visa. As a citizen of South Korea, you don't need a visa for that purpose, and there is certainly no need to "validate" a visa you don't have, nor to validate anything else, by leaving and re-entering the Schengen zone.
The obligation you will have, if you settle with your husband in a country other than Germany, will be to register there if it is required by domestic law. If you want to settle with your husband in Germany then you will need to apply for a residence permit under German immigration law.
- Strictly speaking, free movement applies in EU and EEA countries plus Switzerland. All the EEA countries and Switzerland are in the Schengen area, and it is somewhat more convenient to say "EU and Schengen" than to say "EU and EEA countries and Switzerland."