1) How are you powering the Arduino?
2) Measuring the resistance between the Arduino GND and the +5V pin is NOT a valid measurement or a debugging technique. The fact that two different Arduino boards measure the same would appear to confirm that they are both good.
3) You should be measuring the VOLTAGE not the current between the GND and the +5V pins. When the Arduino is powered (by the undisclosed method) do you see the expected 5V on the +5V pin?
4) When you connect your shield board, does the voltage remain at 5V?
When I connect the shield to the arduino, the resistance between 5V and gnd is very low