By definition an extension of fields $K'/K$ is separable when $K' \otimes_K F$ is reduced for all field extensions $F/K$, and by limit considerations it is the same to say that all finitely generated subextensions are separable. But for finitely generated extensions, separability can be tested via scalar extension against all algebraic extensions (see Theorem 26.2 in Matsumura's *Commutative Ring Theory*), so again by limit considerations it is the same to test using just finite extensions $F/K$. Taking direct limits again in $K'/K$, we see that $K'/K$ is separable if and only if $K' \otimes_K F$ is reduced for all *finite* extensions $F/K$. (See EGA IV$_2$, 4.3.5 and especially 4.6.1 for further discussion on this issue.)

So now consider a normal connected scheme $X$ of finite type over a field $k$ and a non-generic point $x$ of $X$. Let $K$ be the function field of $X$, and $R$ the local ring $O_{X,x}$. The local noetherian completion $\widehat{R}$ is a normal domain by excellence, though for $X$ of dimension 1 (so $R$ is a discrete valuation ring) it is elementary. Let's see that the fraction field $K_x$ of $\widehat{R}$ is separable over $K$ (so applying this to Dedekind $X$ with finite $k$ would settle the question posed).

Letting $F/K$ be a finite extension, we want to show that $K_x \otimes_K F$ is reduced. Let $Y \to X$ be the normalization of $X$ in $F$; if $X = {\rm{Spec}}(A)$ were affine then $Y$ would be ${\rm{Spec}}(B)$ for the integral closure $B$ of $A$ in $F$. Note that $Y \to X$ is finite; indeed, by denominator-chasing we can write $F$ as the fraction field of some $A$-finite subalgebra $B_0$, so $B$ is the integral closure of $B_0$ in its own fraction field, and the $B_0$-finiteness of $B$ is then a special case of the assertion that the normalization of a domain finitely generated over a field is always module finite: see section 33 of Matsumura's *Commutative Ring Theory* for a proof.

Now letting ${\rm{Spec}}(A)$ be an open affine neighborhood of $x$ in $X$, and ${\rm{Spec}}(B)$ its preimage in $Y$, we have $F = K \otimes_A B$, so $$K_x \otimes_K F = K_x \otimes_A B = K_x \otimes_R R'$$
where $R = O_{X,x}$ and $R'$ is the localization of $B$ at the prime ideal of $A$ corresponding to $x$. Since localization preserves reducedness, it suffices to show that $\widehat{R} \otimes_R R'$ is reduced.

By $R$-finiteness of $R'$, the maximal ideals of $R'$ are precisely the primes of $R'$ over the maximal ideal of the local $R$. Thus, $\widehat{R} \otimes_R R'$ is the product of the completions of the semi-local $R'$ at each of its maximal ideals (use Theorem 8.15 in Matsumura's *Commutative Ring Theory* applied to $R'$, whose $\mathfrak{m}_R$-adic topology is the same as its topology for its Jacobson radical). Hence, we just need to show that those completions of $R'$ are reduced. But these are completions of local rings on $Y$, so they're even normal domains (by the same argument used for $X$: excellence in general, or easy considerations with Dedekind domains in the 1-dimensional case).

Local Fields. However, I would just like to point out that for the only positive characteristic global fields, i.e., for finite extensions $K$ of $\mathbb{F}_p(T)$, for every finite extension $L/K$, the integral closure of the DVR $\mathbb{F}_p[T]_{\langle T\rangle}$ in $L$ is finite over the DVR. This is a corollary of the Noether Normalization Theorem. You can find an explicit statement in Eisenbud'sCommutative Algebrasoon after the proof of the Noether Normalization Theorem. $\endgroup$