You don't have to go to Seoul to get your psych meds. There are several English speaking psychiatrists in Cheonan. Ask your regular doctor for a recommendation, or try out Dr. Yu at the Y & K Well-Being Clinic (행복 주는 의원) near the Cheonan Bus Terminal. It's between Starbucks and Krispe Kreme, on the third floor of the building next to Starbucks.
They are confidential, and they give you the medicine right there in the office, so everything is very discrete. (For anyone who knows Dr. Ko at SsangYong Medical, Dr. Yu is her husband.) As usual for Korean clinics, there is no appointment necessary. Just walk in. However, be warned that the nurses don't speak much English.
Depression and anxiety are common among foreigners living in Korea. Don't be ashamed or afraid. Get help if you need it. No one will know, but you will feel much better. (And you can always get a original glazed donut while you're at it.)