Is there an advantage for public school teachers when switching from an E-2 visa to an F-2 visa?

Hello, I am a foreign teacher currently working at a public school in Korea for the third year. Recently, I became eligible to obtain an F-2 visa and am considering switching my visa. I thought that switching to an F-2 visa would allow me to take on other jobs in addition to my main job, but I learned that the public school contract prohibits additional income. Is there any way to modify this contract to allow for part-time work? Also, I am curious if switching to an F-2 visa would have practical benefits if I plan to stay at the public school next year. In particular, I am concerned that my visa points are low, making it difficult to obtain it for more than a year. I would appreciate your advice.