Can I PCS to Korea with my dogs? If so, what is the cost?

Go here for the info you need Capt (Join to see)
I believe the rules will apply to AF as they do to Army.
https://phc.amedd.army.mil/topics/animed/vtfo/Pages/AnimalImport.aspx

Your Base Veterinary Clinic (if you have one) will be able to comply. You need to get started soon on checking the regs and rules. I know from experience as a civilian Vet tech how hard it can be to get pets over to Asia, and often folks find themselves in a bind because they did not research until too late and end up having to quarantine their pets or go through the trouble of having someone back in the US have to take care of the issues and then pay to ship the pet out.

Pay particular attention to the kenneling requirements for pet travel. Depending on the size of your dog, the appropriate travel kennel an be pricey.

Depends sir how long are you stationed there I have heard there is housing for officer at Osan but unless you qualify for housing you don't be able to bring them.

I would wait until you get a sponsor and ask for clarification. You can also contact your gaining organization for guidance.