$A$ and $B$ are Turing-recognizable and their union is $\Sigma^*$, find a decidable $C$ with $A - B \subseteq C$ and $B - A \subseteq \overline{C}$
Got it. The question makes it seem more complicated than it really is.

Let $M$ and $N$ be TMs that recognize $A$ and $B$, respectively.
Construct an enumerator $E$ that behaves as follows:


For each $s_i \in \Sigma^*$ (in lexicographic order):
Run $M$ and $N$ on $s_i$ simultaneously. It must be the case that $s_i$ is in at least one of $A$ or $B$. So, at least one machine is guaranteed to halt.
If $M$ halts, $s_i \in A$, which means certainly $s_i \not\in \overline{A} \cap B$, so print $s_i$.
If $N$ halts, $s_i \in B$, which means certainly $s_i \not\in A \cap \overline{B}$, so do not print $s_i$.


So $E$ prints at least all the strings in $A \cap \overline{B}$, and definitely does not print strings in $\overline{A} \cap B$.
$E$ enumerates a language in lexicographic order, so its language is decidable and satisfies the conditions.