If you can get more time in London, and you're interesting in that sort of thing, I always recommend both the Imperial War Museum and the Tower of London. Both a run on a donation basis, so if you're tight on money you can ponse right on in.
In Edinburgh, I recommend Edinburgh Castle and an underground tour. There's quite a few of 'em around, and I can't remember which one I did, but it was pretty cool. Essentially, the Edinburgh of today is actually built on top of the Edinburgh of hundreds of years ago.
In terms of being socially anxious.... a socially anxious north American is a rockstar by Brit standards. We're a nation of social tards. I'm obviously exaggerating a bit, but culturally, Britain is a much more timid place. Places are less likely to put you on the spot than what could happen over your side of the pond.
EDIT: And just airbnb things. Cheaper than hotels, less socially awkward than couch surfing.
