I’ve seen both of what you describe.
I found that depending of the angle of the spectrograph due to gravity, then there can be small wavelength shifts. Nothing to cause a serious problem during normal observing, though on a run lasting over an hour might then the overall resolution could be slightly reduced. The tests I did were about 3 years ago and I’m not sure I’ve still got a copy. With the spectrograph off the telescope I took neon calibration images with the spectrograph on each side and vertical, to observe the most extreme effects. I then made careful measurements of the centres of the neon lines at the same position on each image.
It might be worth making sure all the screws in the body of the LHIRES III are tight, but I think small shifts are to be expected. There is more than one moving part and heavy cameras attached. Only a minute flexure will cause a small shift in wavelength. That is one reason why fibre fed spectrographs can be more stable, though using optical fibres introduces a different set of problems.
I also see shifts in focus with temperature. So on a cold night if I start with the neon lines in perfect focus then often they won’t be in perfect focus at the end of the run. This is again something I live with. The flat fields will be slightly compromised, but if I refocus then that would also compromise the flat fields. Of course as long as you retake the flats after each refocus then that is not a problem. This is the same behaviour as needing to refocus a telescope due to thermal contraction from falling temperatures.