Tweaks so far:

in collectTweets: 

var targetsheet=ss.getSheetByName("Archive");
...
setArchiveRowsData(targetsheet, data);// write results to Archive sheet



New function:
function setArchiveRowsData(sheet, objects) {
  var headersRange = sheet.getRange(1, 1, 1, sheet.getMaxColumns());
  var firstDataRowIndex = sheet.getLastRow() + 1;
  var headers = normalizeHeaders(headersRange.getValues()[0]);
  var data = [];
  var itemCount=0
  //add things to the archive sheet in chronological order
  for (var i = objects.length-1; i > 0; i--) {
    var values = []
    //we need logic here:
    //if the newly found tweet id is greater than id of last tweet (or timestamp greater?) in archive, add row
    if (true){
      itemCount++;
      for (j = 0; j < headers.length; ++j) {
        var header = headers[j];
        values.push(header.length > 0 && objects[i][header] ? objects[i][header] : "");
      }
      data.push(values);
    }
  }
  if (itemCount>0){
    var destinationRange = sheet.getRange(firstDataRowIndex, headersRange.getColumnIndex(), 
                                        itemCount, headers.length);
    destinationRange.setValues(data);
  }
}