My truck had the shaking/vibration problem. In this huge post, there's been a few theories about the cause of these vibrations and it seems the fixes haven't worked well for all. So I decided to skip over the idea of bringing my truck into the dealership for the part swapping expedition and decided to live with the condition.
If you read carefully, I used the past tense describe the vibration issue. For the past month, I've been using gasoline from a 'top-tier' station (Costco). Initially, there was slightly lesser periods of vibration and the amplitudes were definitely on a smaller scale. For the past two weeks, I drove my truck as my daily driver to see if the vibrations would continue to lessen. Each fill up was at Costco. My daily commute is about 170 miles. Today, my entire trip was vibration free, a first. It was a gradual process but this is my theory: the 5.3L engine must suffer with carbon deposits on the surfaces of the valves (and other components) and using gasoline without the higher levels of detergents found in the top tier gasoline more than likely assists with gunk forming on the valves (and other components). I have just over 5400 miles on my truck and the first vibrations became noticeable around the 1200 mile mark. The other thing I notice is there's more pep to the engine. Pushing down on the gas pedal seems to be met with a smoother engine response and the acceleration doesn't seem to lag. It's quite an improvement on regular 87 octane gasoline.
If you're wondering about 'top tier' gasoline, there's plenty of information found with a basic internet search. There's a few brands offering gasoline that qualifies as 'top tier'.
Lastly, this may be purely coincidental about using a fuel with more detergent and experiencing declining vibrations to the point it has disappeared. I'll keep using Costco gasoline and if those happily forgotten vibrations and shakes return, an update will be made. I lost count on the number of tanks of fuel consumed but it's a big number of refills.