It sounds like a faulty thermostat. However if you look on the back cover can you get the complete model number and I can look it up to see if there could be something else. If this is a 24 volt thermostat check the voltage to the stat and make sure your with in 20-30 volts. Let me know the model when you can.
No the transformer comes off the furnace or air handler. Take cover off the thermostat ( kill power at furnace first) and make sure everything is tight. If you can check voltage that would be great. However this is a standard thermostat and if your getting humming from it I would replace it.
Let me know how it goes.
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