Automatic timestamp when a cell is filled out

You just need to use Apps Script. I'll explain using an example:

function onEdit(e) {
var row = e.range.getRow();
if (row > 1 && e.source.getActiveSheet().getName() === "Sheet1") {
    e.source.getActiveSheet().getRange(row, 14).setValue(new Date());
} else {
    if ((row > 1 && e.source.getActiveSheet().getName() === "Sheet2") || (row > 1 && e.source.getActiveSheet().getName() === "Sheet3")) {
        e.source.getActiveSheet().getRange(row, 6).setValue(new Date());
    }}}

This first of all check which sheet is being edited. If sheet1, then it takes the 14th column of all rows (getRange(row,14)) in that sheet & whenever anything is edited (edit(e)), it adds timestamp (setValue(new Date())) in that 14th column. Similarly, if it's a different sheet,i.e., Sheet2 or Sheet3 or any other name, we can select different columns for timestamp as per requirement. Also, if( (row > 1) condition has been added so that timestamp does NOT get added in the first row upon edit as it's usually headings.

Now you select to get a range of cells & use them as per requirement. See this question for a better idea on getRange().

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());
    }
  }
}

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.