If you need one date, to be replicated throughout a document, follow the instructions at Repeating Data Using Document Property Content Controls and Other Mapped Content Controls and use the Publication Date Document Property (Content Control) as the base.

That gives you a Date Picker Content Control. You can change it anywhere in the document and have the change reflected throughout as soon as you leave the content control. This will work so long as the user is using a Ribbon version of Word (Word 2007 or later on Windows, Word 2011 or later - I think - on the Mac). Content Controls do not work well with mobile devices.