You would want to get a copy of your BA diploma apostilled. To get your diploma apostilled, I will provide you a link for more info: https://www.korvia.com/apostille-korvia-guide/
Yes, you will have to submit a letter of intent to graduate, however, you have to make sure that you have enough time to have your diploma apostilled and have it sent it to Korea.
Your letter of intent to graduate does not need apostille :) you only need to apostille the BA diploma after the graduation (and background check too of course)
please check out Korean consulate web site before you, there is a pretty accurate guide lines for the process. and yes you supposed to get your diploma notarized from public notary before you get them notarized from Korean consulate.