Is this the behavior in an existing document or a new document created from a template?

I would expect it to update in the latter situation.

While I still use CreateDate fields more than I do the DatePicker Content Controls, both act similarly in new documents created from a template in that they update to the current date.

This is just one minor reason to be using templates and creating new documents based on them. There are many more.
Templates in Microsoft Word



By the way, it is a "Content Control" not a field. You will get funny results searching for help on a date picker field.

If this is in new documents created from a template, please write back with more information.