If you'd like to stay in a cottage as opposed to just looking at them, I recommend checking out the National Trust and their self-catering options. https://www.nationaltrust.org.uk/holidays
If you find one in an area that you want to visit, and you'll be car-less just make sure it's within easy distance of stores or restaurants. Having stayed in three of their cottages (two in Cornwall near Polruan and Fowey and one in Robin Hood's Bay), I can say they are excellent accommodations for relaxing and often located in beautiful surroundings.
In terms of areas to base yourself, I'm going to toss out a couple options north of the ones you mention (you didn't say it had to be in the south, so taking a chance here): Kirkby Stephen just outside the lake district, reachable by train. Wonderful small town with green hills and walks right out of town (head up to Nine Standards for great views, stay at Old Croft House for great hospitality). If you can get yourself over to the teeny-tiny village of Keld for a night at Keld Lodge, you will have green hills aplenty and a wonderful path along the river Swale to wander for photos and inspiration (not sure about bus service to Keld though). A larger town with more going on (Castle ruins!), but still very near green open hills that are accessible by foot from the town center, give Richmond (not the London suburb everyone talks about, but the town in Yorkshire) a look. These aren't places generally on the US tourist list, I know them from my two trips on the Coast to Coast path, but they aren't unknown either (walkers/hikers especially will know them).
Wherever you end up, have a wonderful trip and a bit of an adventure!