Let $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d$ follow a multivariate normal distribution. When $\mu = 0, \Sigma = I_d,$ we know that $||X||\sim \chi(d),$ the chi distribution, but I don't see why that'd directly tell me anything about the distribution of $||X||$ in general cases of $\mu, \Sigma?$

If needed, you can assume that $\mu=0$ or $\Sigma$ is diagonal (not identity) or a combination of both. For the moment, I'm more particularly interested in $\mathbb{E}||X||, var[||X||]$. References very much appreciated as well!