Kat,
We registered in the Exeter office. We live in Berkshire, but were getting married in N. Devon. My husband was a trooper, and checked it all out for us. We weren't able to register in Barnstaple (closest registry office to where we were getting married) because they didn't take non-EU people, so they referred us to Exeter.
We knew we had to have 7 or 8 days residency, so we went down on the Fri, stayed the week, and went to the registery office on Mon. We had a note from the people we stayed with to state that we'd been there those days.
The process was really simple. They asked us both questions, but we stayed in the room together. They sent us the registry paper in the post, and we contacted the vicar who was going to perform our service to let her know we had all our paperwork.
I was really nervous, knowing that if we didn't get the registry docs, we couldn't go ahead with our planned wedding. But it all worked out fine. I'm sure you guys will have no problem, and they were really nice there. They are used to having non-EU people (and apparently, have loads of people who don't speak English well, so they love it when Americans come in).
Best of luck!