Character degrees in $2$-blocks of ${\mathfrak{S}}_n$ and ${\mathfrak{A}}_n$

Let $p$ be an odd prime. We show that for sufficiently large $n$, every $2$-block of ${\mathfrak{S}}_n$ and ${\mathfrak{A}}_n$ contains an ordinary irreducible character of degree divisible by $p$. For almost all $2$-blocks of ${\mathfrak{A}}_n$, we classify whether it contains a rational valued ordinary irreducible character of degree divisible by $p$.