Late to the party, but this thread shows up in Google searches. I thought to add another method.
Code:
Sub DropDownValue()
' A scratch macro by Charles Kenyon 29 Sept 2020
' Assigns value of a DropDown Content Control to a string and reports in message box
' https://answers.microsoft.com/en-us/msoffice/forum/all/use-value-not-display-name-of-dropdown-menu-in-vba/0126575d-afa8-457e-aae7-693789f0ba2a
'
Dim ccMyCC As ContentControl
Dim objLe As ContentControlListEntry
Dim strText As String
Dim strValue As String
Set ccMyCC = ActiveDocument.ContentControls(1)
Let strText = ccMyCC.range.Text
For Each objLe In ccMyCC.DropdownListEntries
If objLe.Text = strText Then Let strValue = objLe.Value
Next
MsgBox "The value associated with " & strText & " is " & strValue
Set objLe = Nothing
Set ccMyCC = Nothing
End Sub