Find error on a possibly defective Due

After some successful experiments with a shield my Due started to refuse the serial connection. Before I noticed it's pulling 1 A from the USB port I got the following outputs from journalctl when connecting to the PC:

Programming port:

Mai 29 22:16:22 kai-notebook kernel: usb 1-2: new full-speed USB device number 48 using xhci_hcd
Mai 29 22:16:22 kai-notebook kernel: usb 1-2: New USB device found, idVendor=2341, idProduct=003d
Mai 29 22:16:22 kai-notebook kernel: usb 1-2: New USB device strings: Mfr=1, Product=2, SerialNumber=220
Mai 29 22:16:22 kai-notebook kernel: usb 1-2: Product: Arduino Due Prog. Port
Mai 29 22:16:22 kai-notebook kernel: usb 1-2: Manufacturer: Arduino (www.arduino.cc)
Mai 29 22:16:22 kai-notebook kernel: usb 1-2: SerialNumber: 95635333231351119101
Mai 29 22:16:22 kai-notebook kernel: cdc_acm 1-2:1.0: ttyACM0: USB ACM device
Mai 29 22:16:22 kai-notebook mtp-probe[11703]: checking bus 1, device 48: "/sys/devices/pci0000:00/0000:00:14.0/usb1/1-2"
Mai 29 22:16:22 kai-notebook mtp-probe[11703]: bus: 1, device: 48 was not an MTP device
Mai 29 22:16:22 kai-notebook upowerd[1968]: unhandled action 'bind' on /sys/devices/pci0000:00/0000:00:14.0/usb1/1-2/1-2:1.1
Mai 29 22:16:22 kai-notebook upowerd[1968]: unhandled action 'bind' on /sys/devices/pci0000:00/0000:00:14.0/usb1/1-2/1-2:1.0
Mai 29 22:16:22 kai-notebook upowerd[1968]: unhandled action 'bind' on /sys/devices/pci0000:00/0000:00:14.0/usb1/1-2
Mai 29 22:16:25 kai-notebook kernel: usb 1-2: USB disconnect, device number 48
Mai 29 22:16:25 kai-notebook kernel: cdc_acm 1-2:1.0: failed to set dtr/rts
Mai 29 22:16:25 kai-notebook upowerd[1968]: unhandled action 'unbind' on /sys/devices/pci0000:00/0000:00:14.0/usb1/1-2/1-2:1.1
Mai 29 22:16:25 kai-notebook upowerd[1968]: unhandled action 'unbind' on /sys/devices/pci0000:00/0000:00:14.0/usb1/1-2/1-2:1.0
Mai 29 22:16:25 kai-notebook upowerd[1968]: unhandled action 'unbind' on /sys/devices/pci0000:00/0000:00:14.0/usb1/1-2

Native port

Mai 29 22:15:58 kai-notebook kernel: usb 1-2: new full-speed USB device number 45 using xhci_hcd
Mai 29 22:15:58 kai-notebook kernel: usb 1-2: device descriptor read/64, error -71
Mai 29 22:15:59 kai-notebook kernel: usb 1-2: device descriptor read/64, error -71
Mai 29 22:15:59 kai-notebook kernel: usb 1-2: new full-speed USB device number 46 using xhci_hcd
Mai 29 22:15:59 kai-notebook kernel: usb 1-2: device descriptor read/64, error -71
Mai 29 22:15:59 kai-notebook kernel: usb 1-2: device descriptor read/64, error -71
Mai 29 22:15:59 kai-notebook kernel: usb usb1-port2: attempt power cycle

The 5 V pin has a voltage of 3.5 V and the 3.3 V pin has 2.3 V. The USB however has 5 V. I measured the resistance:
+5V to GND gives me 4.4 kOhm
+3.3V to GND gives me 2.4 Ohm
Can someone please give me the values that are to be expected?

This makes me believe the voltage drops around the polyfuse (which is getting really hot).

I cleaned the board and inspected it with a microscope but to no avail. The SAM3X became extremely hot before I removed the shield.

Any idea how to track this down any further? I don't want to buy a new one before I know what happened and making sure this won't happen to the replacement.