Test on a copy first. See if this does what you want.
Change:
With Worksheets("Database")
lastRowColH = Range("H65536").End(xlUp).Row
Range(Worksheets("Database").Cells(2, 1), Worksheets("Database").Cells(1 + lrow, 1)).Copy
.Range("H" & lastRowColH + 1).PasteSpecial (xlPasteValues)
End With
To this:
Worksheets("Database").Activate
lastRowColH = Range("H65536").End(xlUp).Row
For Each CELL In Range(Worksheets("Database").Cells(2, 1), Worksheets("Database").Cells(1 + lrow, 1))
If CELL.Value <> "" And CELL.Offset(, 1).Value <> "" Then
CELL.Copy
Range("H" & lastRowColH + 1).PasteSpecial (xlPasteValues)
End If
Next CELL
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