I am a bit confused by everyone suggesting Cheltenham. It's a brilliant place to go for the day or a night out - great shopping, cinema, tons of restaurants, loads of nightclubs, big events and festivals, etc. It also has some beautiful architecture and some gorgeous big parks. It's a really lovely place with loads to see and do.
But there's not really any sense of community and it's much, much bigger than all of the places OP lists. Cheltenham has a population of around 120,000 - bearing in mind the large centre with all the shops, clubs, bars, and restaurants. Most of the places that OP has been considering have a population of around 4000-6000. Even Evesham only has a population of 27,000.
Also, as there's such a thriving and large centre in Cheltenham, the population density is much higher than the other places OP mentions which should be another consideration if you're looking for a village-style community.
Cheltenham is absolutely a great place to live with tons to do. But if you're looking for a Cotswold vibe with a local community as the OP says, Cheltenham definitely isn't it!
There are lots of little places surrounding Cheltenham which are more villagey/community type locations and you can nip in when you want. We're about 10-15 minutes outside Cheltenham and it's the best of both worlds.
I'm a Londoner who moved out here to Gloucestershire in 2001 and I've lived in a few different places close to Cheltenham since I've been here. However, while Cheltenham has a lot to offer, it doesn't massively appeal to me because I was looking for something similar to the OP. I think it's because Cheltenham reminds me so much of London, just a smaller version. The architecture, the boutiques, the big green parks (because there are plenty of Cheltenham houses with smaller/courtyard gardens) - for me, having coming from London, I wanted to feel as if I was in Gloucestershire, not a mini-me version of where I'd just left!