I checked the britainusa website's appointment calendar; they have dates open until the 25th of January. So its still going to be a bit before you can book for the 13th of February. Remember, you can only book appointments up to two weeks in advance, so be sure to keep checking, so that you don't miss out on the date that you want.
I know when I applied for my visa, I booked my appointment and plane tickets first, so I knew it was a sure thing. I actually had to shuffle my dates around slightly, as the date I originally wanted wasn't open for whatever reason. I then finished up my online app., paid for the app. fee, and then submitted it. They do ask on the application, where you currently are, when you last visited your fiance', etc. So it probably is best to make the consulate appointment first.
As far as I know, it is perfectly fine to submit your application from the UK whilst you are visiting. If I'm wrong about any of this, Garry or Victoria, please correct me.
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Good luck and let us know how things progress!
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All the best!