Pros and Cons of 3 Different Transducers
LVDT's are very sensitive, but fairly expensive.  They use mutual inductance to determine the position of a moving slug.  You pass a sine wave through an input, and you measure the envelope of the sine wave out to determine displacement.  the PHASE of the output sine wave determines whether you have a positive or negative displacement, so you need phase detection algorithms to provide an output, and the commercial processors tend to be pricey.

Capacitance is defined as 
\$ C  = \frac{\varepsilon \varepsilon_0A}{d} \$, where A is Area of the overlap of the plates of the cap, and d is the distance between the plates.  Thus, the output of a Variable Area device will be linear with area, which you can easily arrange to be linear with displacement.  The output of a separation device is proportional to 1/d, which might be VERY sensitive, but it's not linear.  You can probably build an amp that linearizes it without too much trouble, but, the world not being a perfect place, you'll likely lose sensitivity.  

The circuitry to support either type of capacitive sensor is here.  I lean toward some type of 555 timer arrangement as opposed to any type of AC bridge, which I find tricky.