In several trips to England I still haven't really visited the Cotswolds - Bath is the closest I've come. (Or does the trip to Avebury and Lacock count? It was picturesque as hell, so maybe that was the Cotswolds and I didn't realize it. I was on a London Walks day out and just out to see standing stones and cool things.)
OTOH I had a wonderful time several years ago, when we spent several days in June exploring East Anglia. We had the advantage of traveling by car - driven by a Brit friend, who loved the region and created a great itinerary.
I've just hauled out my scrapbook from that trip. We didn't get up into Norfolk but visited many places in Suffolk and Essex, including Bury St. Edmunds, Lavenham, Kersey, Flatford, Long Melford, Cavendish, Sudbury, Finchingfield, Thaxted, Colchester, Layer Marney, and Greensted.
I don't remember any unpleasant crowds although it was certainly tourist season.
So, please, go and enjoy East Anglia - there's so much to see. But don't go telling everybody about it so that it turns into the Next Big Thing for tourists. Because now that I've looked again through my trip book, I want to see a lot of the region again, and visit places I haven't seen yet, including those in Norfolk. And I hate tourist crowds. Let it just be our little secret. Okay? ;-)