The LVDT is probably alright as it responses as the position changes!
To avoid the generation of negative voltage, one has to tie together all the zero-volt points (digital GND, analog GND, signal GND, PWR Supply GND, and chassis GND); but, practically it is not always possible. There is always some ground loop. Under this circumstances, the sensor is to be calibrated against two known points to accommodate the offset and gain of the sensor. There is no sensor which is ideal; all sensors are real sensors, and they have gain and offset. You may follow these steps to have your sensor calibrated.
1. Apply 15V as excitation voltage.
2. Use a 4.7k + 2.2k voltage divider circuit. Voltage across of 2.2k would be fed to ADC.
3. Keep the position at 10 mm; measure the voltage across 2.2k and record it as VD1. You have the point A(d1, VD1) = A(10, VD1).
4. Keep the position at 40 mm; measure the voltage across 2.2k and record it as VD2. You have the point A(d2, VD1) = A(40, VD1).
5. Take the unknown point C(d, VD).
6. Find the equation for d in terms of VD which is of the form:
d = k*VD + c. //k and c are known constants
7. Use 5V (DEFAULT) as the VREF for the ADC of the UNO. Simple algebraic manipulation will give you:
d = k*(5/1024)*analogRead(A1) + c; //5V --->1023/1024; analogRead(A1) = (1024/5)*VD
8. Declare d as a float variable; also, cast the right hand side of d by (float).