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 $\bar{R}\subseteq C$ 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 $A\subseteq R(A)\subseteq \overline{R(A)}\subseteq C(A)\mathrm{.}$ Then $\bar{A}\subseteq \overline{R(A)}\subseteq C(A)$.

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 $V_s:=\bigcup \{A_t:s(t)=s\}\mathrm{.}$ 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 $\overline{V_s}={\bigcup }_{s(t)=s}\overline{A_t}\subseteq W_s\mathrm{.}$ 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.