I can fully understand why you are unhappy. It seems frustrating to have your English ability questioned.
It sounds to me as if the director genuinely doesn’t understand your accent and is trying to save face by making out it is your problem. Obviously, you are not at fault here. I think it must be very frustrating to work in such an environment and in the long run it will not do your self-confidence any good.
I suggest you try to talk to her again. You should do this in private. Asking in an open space in front of others will not advance your case. If the director wants to save face, she won’t appreciate being challenged in public. Korean culture is very different to British culture and it might help you if you get some advice from a British person who has spent more time in Korea. They might have more of an insight into asking for what you want in that culture. If you really want to keep this job, you might consider eating humble pie.
If this doesn’t succeed, you could think about leaving and taking a job elsewhere. It won’t do you any good to stay there under such circumstances and you are not getting a variety of teaching experiences. However, you should find out about labour law and contractual regulations before taking this route. In Korea, leaving a job is not so straightforward as it is in the UK.