I am a single Staff Sergeant who was promoted on Sept 1st and am currently on a rotation to Korea. Am I entitled to BAH?

Where is your unit housing single SSG now in Korea? in provided housing yes?
You left Hood with no housing costs, so there is no justification to start BAH when you have no housing costs back home to claim. .. BAH is not a pay entitlement based on rank alone (getting promoted to SSG), its is an entitlement based on a situation having met the requirements for that entitlement.. Having housing you are paying for is one of those justifications needed to receive BAH. (And yes we all know the single SSG that backed out of his lease, rented a storage locker for $50 a month, and still receives full BAH while deployed or sering a short tour like you in Korea ...but that SSG HAD housing he was paying for before departing)

I have never seen a SSG live in the B's before but it may be a BN or Post Policy due to OS tours.