Try the following code. Do be careful, though, as it both deletes the content control title and converts the dropdown to a text content control as part of the 'locking' process you wanted. For the output, you'll need a text content control in whichever of the Section's headers you don't have the content control in. The code works equally well with or without a 'different first page' and/or 'different odd and even' page setup.
The other thing to be aware of is that, with content controls in the header, you need to click somewhere else in the header for the macro to run.
Code:
Private Sub Document_ContentControlOnExit(ByVal CCtrl As ContentControl, Cancel As Boolean)
Dim StrDetails As String, HdFt As HeaderFooter
With CCtrl
If .Title = "Sector" Then
If .ShowingPlaceholderText = True Then Exit Sub
.Title = ""
StrDetails = .Range.Text
.LockContentControl = False
.Type = wdContentControlText
For Each HdFt In .Range.Sections(1).Headers
With HdFt
If .Exists Then
If .Range.ContentControls.Count = 1 Then
With .Range.ContentControls(1)
.LockContents = False
.Range.Text = StrDetails
.LockContents = True
.LockContentControl = True
End With
End If
End If
End With
Next
End If
End With
End Sub