The issue you see where the temperature increases when you touch the wires is due to some of the electricity of the circuit flowing through your body. The way the thermocouple works is by passing a small current through it and measuring the voltage drop. If some of that current flows through you and doesn't come back through the TC- wire, it counts as a drop and influences the temperature. You want to make sure that your probes are electrically isolated-- none of the circuit is exposed or uses the metal shield as a conductor as these will allow a path for the current to flow where it shouldn't. You can test this by checking for continuity from the TC+ or TC- wire to either the probe's braid or the tip of the thermocouple element. If it beeps, it's bad (for this circuit).
It may also be helpful to install the noise firmware (AVR Firmware -> Online Repository -> hm-noise) then go to the home screen and wait for the second graph to show up. It should be flat or bouncing between two values. If it is erratic or shows a sine wave then you've got a noise issue which comes from the power supply, power source, or coming in on the probe wires. You can also see a noise reading on the probe by going into the diagnostics display on the device. Navigate the menus to "Manual Mode" then hold the left button to enter the diagnostics display.
You might get a clue about the 5C offset there too. Note that the heat generated on the board influences the measured temperature as well, so if you're testing right after soldering the values will read high until the board returns to room temperature. I usually see a reading 3-4C high when I am testing a board when done soldering. If it stays 5C high even if left sitting overnight, eeeeee not sure where to go after that. Check the 3.3V at the "AVC" pin on the atmega. HeaterMeter expects that to be 3.300V because it is tying to calculate the absolute voltage coming from the amplifier. This can be accounted for if it is too far off by adjusting the mV/C setting int he configuration.