The HDM Module in the fuse box could be the headlight issue, fairly common thing if you search on it.

The shake could be motor mounts or axles. The LS4 eats motor mounts, they are poor design and do not last a super long time. Being that it's an 11 year old car, they are likely to be original depending on mines. I'd look into those first because if the axle geometry is wrong it can cause vibrations.