I don't see why you need such a complicated circuit. A bridge, feeding a differential amplifier...
All you need is a source of current and a buffer or gain stage. Just one resistor, one set of contacts, and one op-amp. (A couple of protection diodes would be a good idea too.)
A simple circuit will not give a linear relationship between skin resistance and output voltage, but neither will your bridge circuit. I would suggest measuring skin conductivity, instead of resistance. In this case you can get a fairly linear relationship between skin conductivity and output voltage if a low voltage range is used across the current shunt resistor, with gain being provided by the op-amp.
The circuit for this is so simple it's scarcely worth drawing up, but...
Current from the positive supply rail flows through the skin and causes a voltage drop across RS. With the value given, and a supply voltage of 10V, a skin resistance of 10 kilohms gives about 1V across RS (actually it's 0.90909V; if you want better linearity, reduce RS and RG in proportion). The op-amp multiplies this voltage by a factor calculated as RF / RG, which in this case is 5, to produce an output voltage proportional to the skin current, which is nearly proportional to skin conductivity (see previous parenthetical comment).
The output voltage therefore is proportional to skin current, which is roughly proportional to skin conductivity, the reciprocal of skin resistance. With no skin touching the contacts, the output will be roughly 0V.
The two diodes are there to protect the op-amp from electrostatic discharge. Extremely unlikely to be a problem with such a low value of shunt resistor; I guess I'm just being thorough.
Any device intended for direct connection to the body must be insulated to the applicable safety standards. I recommend using a fully battery-powered circuit, that cannot be connected to a person while the charger is plugged into it. There may be other legal requirements; it's your obligation, not mine, to determine them and ensure your device complies with them.