MsgBox Appearing Twice

The whole logic is not 100% clear for mre, but anyway, before showing msgbox, you can check what was the target (which dropdown changed) and act accordingly, so for instance:
instead of

        If Range("B3").Value = "OPTION Z" Then

use

        If Range("B3").Value = "OPTION Z" And Target.Address = "$B$2" Then

You need to flip the first to IF test lines. so the first test is the target.address is B2 or B3.

new order:

Private Sub Worksheet_Change(ByVal Target As Range)
Application.ScreenUpdating = False

If Target.Address = "$B$2" Or Target.Address = "$B$3" Then
    If Range("B2").Value <> "" And Range("B3").Value <> "" Then 

<< rest of your code >>

Edit: I noticed for good functioning of the code you also need to move 1 end if statement form 2nd line above the end sub to just above the IF Target.colomn = 2.. tests start
see changes in code below.. I removed the parts with alle the hiding of rows in the select case .. end select section to show the changes in 1 screen

Private Sub Worksheet_Change(ByVal Target As Range)
Application.ScreenUpdating = False
If Target.Address = "$B$2" Or Target.Address = "$B$3" Then
    If Range("B2").Value <> "" And Range("B3").Value <> "" Then 
        If Range("B3").Value = "X" Then
        Rows("8:13").EntireRow.Hidden = True
        Else
        Rows("8:13").EntireRow.Hidden = False
        End If
    End If
    
   Select Case Range("B2").Value
         '<< lot of code deleted better view on changes >>
   End Select
End If ' end if inserted here to end the if test of Target.address = $B$2 

If Target.Column = 2 And Target.Row = 5 Then
    If Target.Value = "Yes" Then
        Rows("15:15").EntireRow.Hidden = False
    Else
       Rows("15:15").EntireRow.Hidden = True
    End If
End If

If Target.Column = 2 And Target.Row = 6 Then
    If Target.Value = "Yes" Then
        Rows("16:34").EntireRow.Hidden = False
    Else
        Rows("16:16").EntireRow.Hidden = True
    End If
End If
' 1 end if was removed here
Application.ScreenUpdating = True
End Sub