Low pass filter vs integrator
Can I use a lowpass filter on the acceleration data instead of an
  integrator and would it eliminate drift?


You probably do want to low pass filter it, but only to eliminate high frequency noise and vibration - not drift. 

Your ultimate goal is to hold a fixed vertical position, but the accelerometer only gives the rate of change in velocity. So to get absolute position you calibrate it for zero acceleration when stationary, then integrate the output to get velocity, and integrate that to get distance (height). 

If the sensor was perfectly accurate and didn't drift at all then the integrated height output wouldn't drift. However in practice there will be some drift, and a small vertical acceleration will still occur. The only practical way to eliminate this is to have an absolute height input, eg. from a ground distance sensor, altimeter (air pressure sensor), or GPS. If you have that then you can high pass filter the integration to eliminate drift from it (but lose long-term height information), then add in low pass filtered absolute position information for total height control.

Note that while double integration makes the height drift very sensitive to accelerometer errors, the acceleration output is rapid so short-term stability can be very good, whereas an absolute height sensor may be too slow and/or inaccurate for this purpose. So for best stability you need both sensor types.