I would say that "first language" is the first language you ever learned, probably the language your parents spoke to you.
While your primary language is the language in which you feel the most confortable expressing yourself. This can change throughout your life: if you stop using your primary language in favour of another one, eventually the new one may become your primary language. Note that a language can be your primary language even if you are not fully (native-like) proficient in it.
For most people they are both the same, and many can't even fathom how can your primary language be a different one from your first language (which is very easy to explain in cases of emigration, like yours).
In your case, your inability to write Hangeul fluidly only means that you are not used to write in Hangeul, has nothing to do with your language proficiency.
So your first language is Korean, and your primary language... well, it may likely be english if you've grown up in an English-speaking environment with limited exposure to and use of Korean, but it is not necessesarily so. So, in which language do you feel more confortable? The answer is your primary language..