Unless I'm missing something, which has been known to happen, you absolutely can adjust the head so it is level front (keypad side) to back (exhaust side). I did it on my machine last week or so and it's very easy to do. Mine was out of whack because the jam nut on the front side was extremely loose.

In a nutshell, you crank it until the back side is level with the notches, remove the tie rod underneath, crank (only front moves) until front is aligned with notches. Reinstall tie rod. You do need to figure out why it's off and correct that issue first. Sounds to me like the poorly installed screw you mentioned allowed the gears to slip.

https://www.carvewright.com/assets/s...level_head.pdf