Keep in mind that the Due is weird in that the USB stack is part of your sketch. The SAM3X processor itself has USB connectivity. There is no external chip providing USB, it is cooked right into the chip and the USB software is running on the chip along with your sketch. Because of this, if your sketch soils the bed then USB will die. This makes the native port potentially difficult when you are testing because you can break the native USB port and then how do you upload a new sketch to fix it? Well, the answer is you erase the chip. With the chip erased it'll run a bootloader from static ROM and that bootloader has a non-corruptible USB stack that will provide a USB port you can upload to.
So, my suggestion would be to erase the chip then upload a very simple sketch - maybe just one that toggles the LED or prints something. Like:
void setup()
{
delay(3000);
}
void loop()
{
SerialUSB.println("It works");
delay(1000);
}
If that sketch gives you a reliable serial port whenever you plug your Due in then the problem was your sketch. If it does not then suspect your USB cord. I don't care if you bought it from the pope or Jesus came down from Heaven and gave you that cord. USB cords are prone to be junk and you should always suspect a bad USB cord if you are having trouble. Seriously, bad USB cords are the root of all evil.