In trying to find alternatives to Scott's paper, I came across Tretkoff Covering Spaces, Subgroup Separability, and the Generalized M. Hall Property, which references this paper of Newman to show surface groups are LERF.
But I don't see how that paper proves it. The main pertinent result of Newman's paper is that for a closed orientable surface of genus $2$, $\pi_1(S)$ embeds in $SL(8,\mathbb{Z})$.
Newman says the results of Scott's paper follow from the following theorem (which I have paraphrased):
Let $F=\{a_1,\ldots,a_k\}\subset \pi_1(S)$, and suppose $b\in\pi_1(S)$ is such that $ba_i\neq a_ib$ for $1\le i\le k$. Then there is a finite index subgroup $H\le \pi_1(S)$ with $b\in H$ but $F\cap H=\emptyset$.
But how does this show $\pi_1(S)$ is LERF? Isn't the conclusion in the wrong direction? By that I mean that LERF would follow if $F\subset H$ and $b\not\in H$. The result as stated seems to only show self-centralizing cyclic subgroups are separable.