Korean immigration rules tend to be very stubborn on requirements of English teachers. Whilst you can get an F2 visa (spouse visa), currently the Ministry of Education requires English teachers to have the below qualifications:
- Minimum Bachelor's degree from a native English speaking country
- Passport from a native English speaking country
This means schools regulated by the public system are subject to these qualifications for their teachers. Therefore, I recommend you apply to Hagwons as they will still look at you.