Engrave text is shaky and out of line

This is almost certainly something mechanical. And it seems quite severe. I would think you should be actually able to see the extraneous motion visually. You may want to take some high speed video of the laser in motion to see if you can identify the source of the movement. If you do this, I suggest mounting the camera rigidly so you’re not adding motion from hand holding.

I’d suggest first checking that the laser module is rigidly mounted to the gantry. Try manually manipulating the laser module in all directions. Do you feel any excess play? If so, identify the source of the play and try to eliminate it.