How does ETS and Terminal Leave work from Korea?

If your command sponsorship is approved already and your spouse is now with you, you should have already extended to meet the SRR for an accompanied tour. The way it is supposed to happen is that your sponsorship is conditionally approved, you extend, and orders are cut. If you don't extend, the orders are rescinded.

On the rare occasion that doesn't happen and you have already received the benefit of command sponsorship, your ETS cannot be extended without your agreement.
Second issue is, you are in your reenlistment window. I don't have my policy book in front of me, but I am certain that you cannot extend only the 6 months needed to meet your DEROS. You will have to either extend 12 months or reenlist 2 years, which will give you almost identical ETS dates right now.

When you come on assignment for your follow on assignment, six months prior to your DEROS, you will sign your declination statement stating that you choose not to extend to have enough time to leave the country. At that point your DEROS will be updated to 3 days prior to your ETS date. As for terminal leave, when you submit your terminal leave, your DEROS will be updated to whatever day you choose to leave the country.

My advice is to extend now to allow yourself the six months to PCS to transition stateside. It is a lot easier to work your next job, residence, etc from a location that's driving distance away and on the same time zone.