Moment of Inertia of rocket

I am trying to model a rocket launch, and am trying to calculate its moment of inertia at each instant. I am assuming a two-stage rocket with a liquid fuel tank, plus a payload (nosecone-shaped for example), so the configuration is something like a hollow cylinder with an inner cylinder that decreases in height with time.

Therefore I am wondering, will the total moment of inertia just be the sum of the cylinder and hollow cylinder of each stage? Also, since the centre of mass is constantly changing, will this affect how I calculate the moment of inertia about the CoM?

When I use values taken from the Falcon 9, I get values like $1.25*10^6$. Is this reasonable? I can't seem to find any reference values online to check my answers against…

Thank you in advance!