Hey, I'm not sure if this is still relevant, but as another American who sort of had to move here sight unseen, I would say that AirBnB is your best friend while you look for a flat. When I was looking in Sheffield I was paying about £20 per night, so you don't have to rush due to a hotel budget or something.
See if your uni has any help with private accommodation - my uni at undergrad has a site that only lists landlords and properties they've vetted,so you know it's not a scam, and they don't charge agency fees.
I found that letting as a non-EU student is really difficult with letting agencies, they will ask for 6 months up front - from everyone in your property (maybe not such a big deal if it's only a flat for the two of you, but still a lot of money at once!). With private landlords you tend to get a lot more flexibility.
And from my experience background checks aren't really a thing, they'll just ask for a scan of your passports and proof that you're a student.

I also found that my Master's is going a lot better than my undergrad - I barely got a 2:2 and I failed my dissertation, but I'm consistently getting 70+ at post grad, with what I perceive to be *less* work. I think a large part for me is that I like my current subject a lot better, but I agree that it's also already having experience in research and academic stuff.