I have very limited knowledge of this, but can offer the info that when my daughter had to get a visa from the US Embassy in London, despite the website saying the appointments were available within 2 weeks, it was about a 6 week wait for an appointment. Telling the official on the phone that she was in total contradiction to their own web site did nothing to help, as I'm sure you can all imagine. Only option was to book appointment at Belfast Embassy, which has shorter waiting list. So, give yourself plenty of time to get this sorted.
Oh, and be prepared for a big phone bill! Making the appointment runs up a bill of about 6 quid.

Best of luck, BlackCat. Let us know how you get on.