Can I stop the batteries in a device from draining by taking out just one?
You are correct that removing one of a series connected set of cells will break the circuit.
More than likely the unit goes into deep sleep mode when off so it is not worth the effort. If you have access to a multimeter you can set it to the lowest mA range and measure the current while it is asleep by connecting the meter in series with the cells somehow. If you find the datasheet for the batteries you will get a figure for capacity in mAh at a certain drainage rate. Divide this by your current reading to estimate the number of hours.
$$ h = \frac {mAh}{mA} $$
You will get far more than this because the standby current is not stressing the chemistry.