1) You can create a method in the AnotherDatePicker class
like this:
Public Sub getTimeStamp As String
Return dateField.Text
End Sub
And then you can get the string using:
Sub AnotherDatePicker1_DateChanged (NewDate As Long)
Log ($ "New date $ Date {NewDate}" $)
Log (AnotherDatePicker1.TimeStamp)
End Sub
2) (I guess) StateManager saves views values. AnotherDatePicker is a class. It is not a View.
However, you could save and retrieve the value using a invisible label in your activity.