Q. In running text, what is the proper way to write out a URL—with the www or without?

A. Please see CMOS 8.191 (“Titles of Websites and Web Pages”):

Many websites either do not have a formal title or do not have a title that distinguishes it as a website. These can usually be identified according to the entity responsible for the site along with a description of the site and, in some cases, a short form of the URL. For example, http://www.apple.com/ might be referred to in running text as Apple.com.

Browsing in the Manual, you might also notice that in running text we have chosen to include the entire protocol, beginning with http (CMOS 6.8, 7.46, etc.).