Hello Ricker090,

I changed the macro to start at "A7" on the individual destination sheets. The macro has been tested and works. Replace your old macro with this code.
Sub MoveData_1a()

  Dim DataTitle As String
  Dim DstRng As Range
  Dim DstWks As Worksheet
  Dim Headers As Range
  Dim NextRow As Range
  Dim R As Long
  Dim Rng As Range
  Dim RngEnd As Range
  Dim SrcWks As Worksheet
  
    Set SrcWks = Worksheets("Master Data")
    
    Set Headers = SrcWks.UsedRange.Rows(1)
    
    Set Rng = Headers.Offset(1, 0)
    Set RngEnd = SrcWks.Cells(Rows.Count, Rng.Column).End(xlUp)
    Set Rng = SrcWks.Range(Rng, RngEnd)
    
    Application.ScreenUpdating = False
    
      For R = 1 To Rng.Rows.Count
        DataTitle = Rng.Item(R, 1) & Rng.Item(R, 2) & Rng.Item(R, 3)
        
        If DataTitle <> SrcWks.Name Then
           On Error Resume Next
           Set DstWks = Worksheets(DataTitle)
             If Err <> 0 Then
                Worksheets.Add After:=Worksheets(Worksheets.Count)
                ActiveSheet.Name = DataTitle
                Set DstWks = ActiveSheet
                Err.Clear
             End If
           On Error GoTo 0
           
           Set DstRng = DstWks.Range("A7")
           Set NextRow = DstWks.Cells(Rows.Count, "A").End(xlUp)
           
             If NextRow.Row < DstRng.Row Then
                Set NextRow = DstWks.Range("A7")
                Headers.Copy NextRow
             End If
           
           Set NextRow = NextRow.Offset(1, 0)
           Rng.Rows(R).Copy NextRow
        End If
      Next R
    
    Application.ScreenUpdating = True

End Sub