Ok, as a Brit I'll add to the advice on not coming across as obnoxious Americans (though in doing so, I will come across as an obnoxious Brit, sorry. I'll say this now otherwise it will be few tips and lots of disclaimers. I hope I don't offend anyone):
I second on speaking more quietly. That's probably the main one. Also, waiting patiently. You don't need that drink right now, the service isn't 'terrible' if the waiter is busy dealing with someone else. (Wages for service staff are high here. Everyone is paid the minimum wage and tips are smaller. As a result businesses can't flood the floor with staff as we have found elsewhere).
The food isn't bland, it just isn't full of sugar. (I cannot believe the things US Supermarkets put sugar in.) Also, we seem to like our food a lot drier, there are fewer sauces. In a pub, you can usually just ask for more gravy or ketchup or whatever, and they will be happy to oblige. You will find eating out insanely expensive. And the portions small. Avoid soft drinks. They are tiny compared to what you are used to, pricey and no-one has ever heard of a free refill.
You will probably consider us stand-off-ish. Sorry about that. We will probably consider you fake, sorry. No-one will tell you to 'Have a nice day' unless they actually mean it (ie no-one will ever say it. If you are friends, it is implied that they hope you have a nice day. If you are paying them for a service, it will seem, I dunno, like you are paying them to pander to you, which is not appreciated).
Also, people will say 'thank you', but never 'you're welcome'. We just don't have the equivalent to danke and bitte, it's just 'thanks' here. We're not being rude, it's just a language thing.
Don't tell me I have a 'cute' accent. It's patronising. And you are the one with the accent when we're on British soil. (By all means say you like the British accent, just avoid 'cute').
I also agree with the sneakers thing, nothing says American tourist like bright white running shoes. I really don't get it either. Us Europeans walk a lot, and we just wear normal leather shoes, and we are fine. When you are just walking around a city, sneakers, waterproofs etc is just overkill. Also, 'bum bags'. I think you call them 'f*nny packs'. That means something totally different here (seriously don't say f*nny)! And visors. The sun is not strong enough to ever necessitate them.
Don't tell me about your torn rotator cuff, your IBS, your allergies, your intolerance to whatever. Get on with things quietly. I'm sorry you have a bad knee or whatever, but your 'surgeries' are your business. Any kind of bracing device will make us think you are 1) neurotic about your health 2) suing someone for something 3) getting scammed by a doctor who is making a huge profit. (I know none of this may be true. I'm just saying, this is what the Brits are thinking)
Also, don't tell me about the medications you're on. Or mention how much we drink. You think we drink too much, we think you take too many pills. We are probably all achieving the same thing.
On the same subject, NEVER be rude about the NHS. By all means ask how it works, what we think of it, how we would improve it. But seriously, don't diss it. We love our NHS with the fire of a thousand suns.
Don't mention our teeth. We know. We are sensitive about it.
My biggest tip is just don't make a fuss. Honestly, 'Keep calm and carry on' is a slogan for a reason. It really isn't as bad as you think, you'll be fine. (this applies to public transport, hotel rooms, service, food. If you don't want to be seen as obnoxious just remember 'don't make a fuss'.)
All that said, the fact you are asking how not to be obnoxious tells me you're not obnoxious. So you can probably ignore all of the above and just carry on being a decent person.
As for where to stay, Cambridge is a great suggestion, and there are lots of pretty villages nearby. Norwich is also beautiful, lots of villages, coastline, and cheaper. About 2 hours from London on a very regular train service.
As for Churches, the Church of England (local parish churches) are usually stunning buildings. Particularly in Norfolk, they will most by Norman (so built after the Norman conquest of 1066). However, the congregation of the C of E are definitely at the more standoffish end of the spectrum. They are the upright citizens of the village, it will take months to get much more than a polite, 'How do you do?'. If you are looking for a more social experience, I would suggest the local Methodist church. They tend (in my experience) to be more welcoming, and have more mixer type events. That said, they often don't drink. Ugh.
You will find living in a village difficult without a car. Buses are infrequent, and taxis are expensive. If you do decide to drive, read up on roundabouts.
And enjoy yourselves! What a great trip!
PS feel free to give me tips on how to be a better tourist in the US. We love to visit! (And yes, I say, 'It's all so big!' to everyone I meet. Is that endearing or annoying?)