As a US citizen you can travel to Korea without a visa for a stay of up to 90 days and can get married during this time. The Seoul embassy in the US has published some useful guidelines on this here. However, to change to a spouse visa, you will need to leave the country.
As far as I can tell, Korea doesn't allow its citizens dual citizenship (in other words your husband-to-be would have to renounce Korean citizenship to become a US citizen). In recent years the law has also changed so that makes finding accurate information online difficult. Ultimately the laws regarding this are quite complex, and there are exceptions, so I would advise consulting your local embassy on this issue.
Best of luck.