We stayed in Chipping Campden at the Volunteer Inn, which was inexpensive and great. It had the best pub in town and a great Indian restaurant.
I am not impressed with this option. I researched this problem before we did 4 weeks driving in Wales and England in October 2017. You will waste a lot of time trying to connect with transport, which will largely be by bus.
Driving on the left is not as bad as you might think, you just have to concentrate, take your time and have someone reminding you to stay on the left. British drivers are not aggressive or rude, which helps.
If you still don't want to rent a car, consider taking a guided tour.
You can see the Cotswolds thoroughly in two full days. Also, consider visiting Bath, Oxford, Stratford Upon Avon and Blenheim Palace.