Here's a way of thinking about it that I find helpful.
A key intuition behind Many-Worlds is that the laws of quantum mechanics are universal. That is to say, the descriptions we apply to electrons, buckyballs, cats, observers, and the Earth are all fundamentally the same. They are "merely" quantum systems of increasing size.
Therefore, we can ask ourselves simpler versions of the energy question. A question like "how is the energy different when Earth goes into a superposition?" certainly seems confusing. But the answer to that is fundamentally the same as the answer to "how is the energy distributed when Schrödinger's cat goes into a superposition?" And if that's confusing, we may even ask "how is the energy distributed when an electron goes into a superposition?"
I'll answer the question from the perspective of Schrödinger's cat. Suppose we put a cat and a radioactive atom into a perfectly isolated box. After some time, the atom will decay, triggering a detector which releases sleeping gas. (I like to keep the cat alive in both branches.)
Write ##\ket{C}## for the various states of the cat, write ##\ket{A}## as the state of the atom before it has decayed, and write ##\ket{D}## as the state of the atom after it has decayed. I'll also write ##\ket{\psi}## as the state of the entire system. The time evolution then looks something like
$$
\begin{aligned}
\ket{\psi} =&\, \ket{A} \ket{C_\text{awake}} \\
\rightarrow&\, \frac{1}{\sqrt 2} (\ket A + \ket D) \ket{C_\text{awake}} \quad \text{(atom decays)} \\
= &\, \frac{1}{\sqrt 2} \ket A \ket{C_\text{awake}} + \frac{1}{\sqrt 2} \ket D \ket{C_\text{awake}} \\
\rightarrow &\, \frac{1}{\sqrt 2} \ket A \ket{C_\text{awake}} + \frac{1}{\sqrt 2} \ket D \ket{C_\text{asleep}} \quad \text{(cat interacts with atom)}
\end{aligned}
$$ So then, after the time evolution we have two cats, one asleep and one awake. This is analogous to the situation where an experimenter measures, say, a qubit, causing the wavefunction to branch into two worlds.
We can then ask, "how is the energy distributed between the awake-branch and the asleep-branch?" To begin, note that the individual states ##\ket A \ket{C_\text{awake}}## and ##\ket D \ket{C_\text{asleep}}## contain (at least to a very good approximation) the same energy. (The main contribution to this is the rest mass of the cat, which is of course the same in both cases). We can express this using the Hamiltonian ##\hat H## for the entire system:
$$
\begin{aligned}
\hat H \ket{A} \ket{C_\text{awake}} \simeq E \ket{A} \ket{C_\text{awake}} \\[0.8em]
\hat H \ket{D} \ket{C_\text{asleep}} \simeq E \ket{D} \ket{C_\text{asleep}} \\
\end{aligned}
$$ Then, notice what happens when we have a superposition of the asleep-branch and awake-branch. Applying the Hamiltonian operator yields
$$
\begin{aligned}
&\hat H \left( \frac{1}{\sqrt 2} \ket{A} \ket{C_\text{awake}} + \frac{1}{\sqrt 2} \ket{D} \ket{C_\text{asleep}} \right) \\
&= \frac{1}{\sqrt 2} \big( \hat H \ket{A} \ket{C_\text{awake}} \big) + \frac{1}{\sqrt 2} \big( \hat H \ket{D} \ket{C_\text{asleep}} \big) \quad \text{(by linearity)} \\
&\simeq \frac{1}{\sqrt 2} \big( E \ket{A} \ket{C_\text{awake}} \big) + \frac{1}{\sqrt 2} \big( E \ket{D} \ket{C_\text{asleep}} \big) \\
&= E \left( \frac{1}{\sqrt 2} \ket{A} \ket{C_\text{awake}} + \frac{1}{\sqrt 2} \ket{D} \ket{C_\text{asleep}} \right)
\end{aligned}
$$ As we can see, the eigenvalue of the superposition under ##\hat H## is also ##E##. That is to say, the energy of the superposition is the same as the energy of the individual parts. (Note that this has nothing to do with the factors of ##1/\sqrt 2##; those are just there for normalization.)
How can we interpret this? Think about how this must look from the cat's perspective. It doesn't suddenly feel its mass half in two, it either just feels some sleeping gas getting released, or it keeps doing whatever it was doing before. In the language of quantum mechanics, then, I would say that the energy is shared between the asleep-branch and awake-branch of the wavefunction.
And because there's no fundamental difference between superpositions of cats and superpositions of Earths, the same argument applies whenever an observer measures, say, a qubit.
So that's my answer to the question. I think the confusion arises from thinking of energy as something that exists inside each world - this is the classical intuition that we are used to. But in Many-Worlds, the universe fundamentally lives in Hilbert space, not Euclidean space. And in Hilbert space, multiple worlds can share the same energy.