$2^\kappa$ for uncountable $\kappa$ is a ZFC example of an almost sequential non-sequential space. To see why it's almost sequential, let $x$ be any point in $2^\kappa$ and let $D=\{y \in 2^\kappa: |\{\alpha < \kappa: x(\alpha) \neq y(\alpha) \}| < \aleph_0 \}$. Then $D$ is a sequential (even Fréchet-Urysohn) dense subspace of $2^\kappa$ which contains $x$.

$2^\kappa$ for uncountable $\kappa$ is a ZFC example of an almost sequential non-sequential space.

The easiest way to see why it's not sequential is to note that there is a set $A \subset 2^\kappa$ and a point $p \in \overline{A}$ such that $p$ is not in the closure of any countable subset of $A$. It suffices to take $A=\{x \in 2^\kappa: |x^{-1}(1)| < \aleph_0 \}$ and $p \in 2^\kappa$ to be the function constantly equal to 1.

To see why it's almost sequential, let $x$ be any point in $2^\kappa$ and let $D=\{y \in 2^\kappa: |\{\alpha < \kappa: x(\alpha) \neq y(\alpha) \}| < \aleph_0 \}$. Then $D$ is a sequential (even Fréchet-Urysohn) dense subspace of $2^\kappa$ which contains $x$.