The following code has worked good for me in the past:

Dim rng As Range
Set rng = ActiveSheet.AutoFilter.Range
rng.Copy Destination:=Worksheets("Old Data").Range("A1")
Sheets("Old Data").Range("A1").EntireRow.Delete Shift:=xlUp

However, I now want it to "Insert the Copied Data" at the destination and
shift the existing data down.

The code currently allows the data to be pasted over any existing data???