Dataset Copy, Change, then Write new Rows back
So basically you make a copy of all rows for a specific job, replace the job number and append the changed row back to the database. Sounds like this should be doable in one loop with the help of ItemArray and no need for a temporary table:

nexusDBDataSet.Drawing_Table.Clear();
drawing_TableTableAdapter.Fill(nexusDBDataSet.Drawing_Table);
foreach (DataRow jobRow in nexusDBDataSet.Drawing_Table.Select(string.Format("JobNumber = '{0}'", OldJobcardNumber)))
{
    var newRow = nexusDBDataSet.Drawing_Table.NewRow();
    newRow.ItemArray = jobRow.ItemArray;

    string directoryName = jobRow["DirectoryName"].ToString();
    newRow["JobNumber"] = NewJobcardNumber;
    newRow["DirectoryName"] = DirectoryName.Replace("\\" + OldJobcardNumber + "\\", "\\" + NewJobcardNumber + "\\");

    nexusDBDataSet.Drawing_Table.Rows.Add(newRow);
}
// save table