Under choice, every open cover of a paracompact Hausdorff space has locally finite open refinements $\{V_s\}$ and $\{W_s\}$ with $\overline{V_s}\subseteq W_s\subseteq\overline{W_s}\subseteq U_s$

We first prove a one-shrink construction for any open cover $\mathcal C$. Let $\mathcal R$ be the family of all open $R$ for which $\overline R\subseteq C$ for some $C\in\mathcal C$. By [L1] and [L3], $\mathcal R$ covers $X$. Take a locally finite open refining cover $\mathcal A$ of $\mathcal R$ by [F1], discard its empty members, and use [A1] to assign to each $A\in\mathcal A$ sets $R(A)\in\mathcal R$ and $C(A)\in\mathcal C$ with $A\subseteq R(A)\subseteq\overline{R(A)}\subseteq C(A).$ Then $\overline A\subseteq\overline{R(A)}\subseteq C(A)$.

Apply step 1.1 to $\mathcal U$. This gives a locally finite open cover $\{W_s\}_{s\in S}$ and assigned $U_s\in\mathcal U$ such that $\overline{W_s}\subseteq U_s$.

Apply step 1.1 again, now to the cover $\{W_s:s\in S\}$. Obtain a locally finite open cover $\{A_t\}_{t\in T}$ and a map $t\mapsto s(t)$ such that $\overline{A_t}\subseteq W_{s(t)}$. For $s\in S$ put $V_s:=\bigcup\{A_t:s(t)=s\}.$ The family $\{V_s\}_{s\in S}$ is an open cover. It is locally finite because any neighbourhood meeting only finitely many $A_t$ meets only the corresponding finitely many grouped unions $V_s$.

Each subfamily $\{A_t:s(t)=s\}$ is locally finite, so [L2] gives $\overline{V_s} =\bigcup_{s(t)=s}\overline{A_t}\subseteq W_s.$ Together with step 2.1 this yields $\overline{V_s}\subseteq W_s\subseteq\overline{W_s}\subseteq U_s$ for every $s$, with both displayed families locally finite open covers.