Automatic timestamp when a cell is filled out

Stackoverflow is a place to ask questions related to programming, e.g. that you're actually working on. Not really asking for others to develop it for you, i.e. you didn't even started trying any Apps Script code yet. I recommend you reading its tutorials and guides. It's really easy to start.

Anyway, just to help you get started, I'll drop everything you said and stick to the question title: "automatic timestamp when a cell is filled out"

I advise you to do it all on apps script, and drop your formulas entirely, e.g.

function onEdit() {
  var s = SpreadsheetApp.getActiveSheet();
  if( s.getName() == "Sheet1" ) { //checks that we're on the correct sheet
    var r = s.getActiveCell();
    if( r.getColumn() == 4 ) { //checks the column
      var nextCell = r.offset(0, 1);
      if( nextCell.getValue() === '' ) //is empty?
        nextCell.setValue(new Date());
    }
  }
}

This code does what I understood from yours, which is: if something is edited on column D and column E is empty, add the current date to E.

Just addition to above code FOR Multi Column AutoStamp in Same Sheet

function onEdit() {
  var s = SpreadsheetApp.getActiveSheet();
  if( s.getName() == "Sheet1" ) { //checks that we're on the correct sheet
    var r = s.getActiveCell();
    if( r.getColumn() == 5 ) { //checks the column
      var nextCell = r.offset(0, 1);
      //if( nextCell.getValue() !== '' ) //is empty?
      nextCell.setValue(new Date());
    }

    if( r.getColumn() == 7 ) { //checks the column
      var nextCell = r.offset(0, 1);
      //if( nextCell.getValue() !== '' ) //is empty?
      nextCell.setValue(new Date());
    }

    if( r.getColumn() == 9 ) { //checks the column
      var nextCell = r.offset(0, 1);
      //if( nextCell.getValue() !== '' ) //is empty?
      nextCell.setValue(new Date());
    }
  }
}