In the After Update event of your date field text box put this code:
Me![Date].Tag = Me![Date].Value
Then, in the On Enter event of the the date field text box put htis code:
If Not (IsNull(Me![Date].Tag) Or Me![Date].Tag = "") Then Me![Date].Value = me![Department].Tag
End If
Make sure you change [Date] to whatever your date field name is. This should cause the date to carry over to the next form until you close the form.
(Thanks to Jack Cowley for the code.)