*winces* No, posh is not an acronym. That particular tale has been debunked so many times (like Ship High In Transit or For Unlawful Carnal Knowledge), I'm always surprised it continues to survive. If you want to know the origin of a word (look it up, frankly!) always be suspicious of an acronymic explanation; they're mostly backronyms. Acronyms are a very modern obsession; they really only began to be used as shorthand from about WWI onwards (Ack-ack, Toc H are early versions).

Posh most likely derives from Romany, although it's not certain (the OED gives a more nebulous origin, for example). According to Partridge's dikker of slang, posh is money, specifically a low value coin such as a halfpenny, and reckons it's recorded as early as the mid 19thC. It gives a second definition of "a dandy", which is the OED origin, but that only dates to the end of the 19thC. Essentially, if you'd got a bit of "posh", you'd got a bit of money. "Port out..." was first recroded only as late as 1935, and it wasn't associated with P&O until 1937 (the '35 mention claimed it to be a US shipping term).

As for posh folk singing folk songs, surely the key word here is folk? Posh folk are folk too, so why shouldn't they sing folk! ;-)