For $q > 1$ we know that $\xi$ dies in $H^q(X_{\overline{K}}, (X_{\overline{K}} \to \Spec(K))^{-1}\mathcal{F})$ (Theorem \ref{theorem-vanishing-affine-curves}). By Lemma \ref{lemma-directed-colimit-cohomology} we see that this means there is a finite extension $K'/K$ such that $\xi$ dies in $H^q(X_{K'}, (X_{K'} \to \Spec(K))^{-1}\mathcal{F})$. Thus we can take $k' = k$ and $Z = X$ in this case. \medskip\noindent Assume $q = 1$. Recall that $\mathcal{F}$ corresponds to a discrete module $M$ with continuous $\text{Gal}_K$-action, see Lemma \ref{lemma-equivalence-abelian-sheaves-point}. Since $M$ is $n$-torsion, it is the uninon of finite $\text{Gal}_K$-stable subgroups. Thus we reduce to the case where $M$ is a finite abelian group annihilated by $n$, see Lemma \ref{lemma-colimit}. After replacing $K$ by a finite extension we may assume that the action of $\text{Gal}_K$ on $M$ is trivial. Thus we may assume $\mathcal{F} = \underline{M}$ is the constant sheaf with value a finite abelian group $M$ annihilated by $n$. \medskip\noindent We can write $M$ as a direct sum of cyclic groups. Any two finite \'etale Galois coverings whose Galois groups have order invertible in $k$, can be dominated by a third one whose Galois group has order invertible in $k$ (Fundamental Groups, Section \ref{pione-section-finite-etale-under-galois}). Thus it suffices to prove the lemma when $M = \mathbf{Z}/d\mathbf{Z}$ where $d | n$. \medskip\noindent Assume $M = \mathbf{Z}/d\mathbf{Z}$ where $d | n$. In this case $\overline{\xi} = \xi|_{X_{\overline{K}}}$ is an element of $$ H^1(X_{\overline{k}}, \mathbf{Z}/d\mathbf{Z}) = H^1(X_{\overline{K}}, \mathbf{Z}/d\mathbf{Z}) $$ See Theorem \ref{theorem-vanishing-affine-curves}. This group classifies $\mathbf{Z}/d\mathbf{Z}$-torsors, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-torsors-h1}. The torsor corresponding to $\overline{\xi}$ (viewed as a sheaf on $X_{\overline{k}, \etale}$) in turn gives rise to a finite \'etale morphism $T \to X_{\overline{k}}$ endowed an action of $\mathbf{Z}/d\mathbf{Z}$ transitive on the fibre of $T$ over $x_{\overline{k}}$, see Lemma \ref{lemma-characterize-finite-locally-constant}. Choose a connected component $T' \subset T$ (if $\overline{\xi}$ has order $d$, then $T$ is already connected). Then $T' \to X_{\overline{k}}$ is a finite \'etale Galois cover whose Galois group is a subgroup $G \subset \mathbf{Z}/d\mathbf{Z}$ (small detail omitted). Moreover the element $\overline{\xi}$ maps to zero under the map $H^1(X_{\overline{k}}, \mathbf{Z}/d\mathbf{Z}) \to H^1(T', \mathbf{Z}/d\mathbf{Z})$ as this is one of the defining properties of $T$. \medskip\noindent Next, we use a limit argument to choose a finite extension $k'/k$ contained in $\overline{k}$ such that $T' \to X_{\overline{k}}$ descends to a finite \'etale Galois cover $Z \to X_{k'}$ with group $G$. See Limits, Lemmas \ref{limits-lemma-descend-finite-presentation}, \ref{limits-lemma-descend-finite-finite-presentation}, and \ref{limits-lemma-descend-etale}. After increasing $k'$ we may assume that $Z$ splits over $x_{k'}$. The image of $\xi$ in $H^1(Z_{\overline{K}}, \mathbf{Z}/d\mathbf{Z})$ is zero by construction. Thus by Lemma \ref{lemma-directed-colimit-cohomology} we can find a finite subextension $\overline{K}/K'/K$ containing $k'$ such that $\xi$ dies in $H^1(Z_{K'}, \mathbf{Z}/d\mathbf{Z})$ and this finishes the proof.