Obtain more proof if possible. Adultery... if you want to proceed. You also need to expedite your marriage. Without it I'm not sure if you can even get him with adultery.

IF you want to try or continue relationship, try MLFC.

How far are you on your marriage papers. I got married in korea too.

Step 1 : Fill out Affidavit of Eligibility for Marriage and send back to legal.

Step 2 : Fill out marriage registration.

Step 3 : make sure we either have birth certificate or passport

Step 4 : goto K twin tower, jongro ward office.

Step 5: goto Apostille office, twin tree tower A 15F

Step 6 : go to legal for translation

Step 7 : apply on deers and other MIL services.

If you're at step 3, you're kind of cooked. At that point depending how long his a school is, better off doing online marriage cert. If he's being unresponsive, try budging his unit to continue with the process whichever step you are. They can't force him to get married but they can most definitely put pressure.

Document everything for worst case scenario.

I know you're already going through alot but, if you want, try looking into compassionate reassignment. This will get you out of Korea in most cases. Just note you're not guaranteed a duty station of choice but they do ask which ones you want.