Voltage Drift on an Open Circuit (and how to get rid of it)
The slow drift may be due to leakage currents in your sensor.  Since the input is open-circuit (theoretically infinite impedance), even a tiny leakage current can cause a significant offset voltage.

The simplest solution is to put some resistance across the sensor.  1 M&Omega; will likely swamp the leakage signal.  10 M&Omega; may work too.  It depends on the magnitude of the leakage and the input impedance of whatever is receiving the raw signal.

You want to make the resistance low enough to bring the offset you are seeing down to don't-care levels, but not so low as to significantly load the mouse-tongue-on-metal battery.  I suspect the latter has significantly lower impedance than 1 M&Omega;, in which case that's a good value to use.