This is one way you could do it :
var strings:Array = ["one", "two", "three", "four", "five", "six"];
// create a clone of your array that represents available strings
var available:Array = strings.slice();
// choose/splice a random element from available until there are none remaining
while (available.length > 0)
{
var choiceIndex:int = Math.random() * available.length;
var choice:String = available[choiceIndex];
available.splice(choiceIndex,1);
trace (choice);
// you could create a textfield here and assign choice to it
// then add it to the display list
}
The concept is that you create a clone of the array and then randomly take 1 element from that array, until you have none left. That ensures that you never have a duplicate.