The geniuses who wrote Word came up with two types of fields that are updated automatically.
Some are updated when you call up your document.
Some, including the date field, are updated only when you print or preview your document.
To make sure that the fields are actually updated when you print, go to Tools/Options.
In Options, click on the Print tab, then click on the "Update fields" box to place a check mark in it. If not seeing the correct date on a document when you are working on it really drives you batty, right-click on the date to highlight it. In the pop-up menu, choose "Update field."