We are doing a big redesign work on our website, and at least 50% of the website folders will be moved to new places with different names (i.e. many URL will be changed). Sure, Google will need some time to index new pages and we expect our SERP positions will be not so good as they are now for some time.

We also have an old idea to remove www from our domain name. It seems, it's the right time to do these two works together with publishing the website with updated contents. Or is it better from SEO perspective first publish the new contents, and only after some time, when our SERP positions will return to prior results, tell Google that the domain name without www is our preferred domain name?