I check, or average out, mechanical shifts in my spectrograph by taking neon calibration frames at the start and end of each stellar spectra. However, I’m using an LHIRES III which is more prone to flexure than the Alpy, plus I frequently stay on the same star for an hour or more. So you may not need to do this for your setup.
When checking a radial velocity, then ISIS has a useful tool in the “Misc” tab for calculating the heliocentric velocity correction, so the rotation of the Earth and its orbit around the Sun (bigger contribution) can be calculated and removed. Then I find SIMBAD is an excellent resource for checking the expected radial velocity of the star.