Well, I stayed on hold for two hours and finally got through! In that time I tried (and failed) to use their online contact form, posted in a Facebook group, posted on here, tweeted Princess (and got a reply saying I would need to phone), read the advice I got in the Facebook group, and booked direct with the TA.
I now have a deluxe balcony cabin booked for £500 less than I was paying for a regular balcony, and have $200 extra on board spend.
@dides - by phoning in the evening and going through to a US agent, you also just get hold music, rather than the really irritating cheesy adverts while on hold to Princess UK.