Let $(X,\mathscr{O}_X)$ be a scheme and $\mathscr{F}$ be an $\mathscr{O}_X$-module. It can be shown that $\mathscr{F}$ is quasicoherent iff for every affine open $U = \operatorname{Spec} A$ and $s\in A$, the canonical morphism $$ \Gamma(U,\mathscr{F})_s\to \Gamma(D(s),\mathscr{F}) $$ is an isomorphism. In particular, when $X$ is qcqs, by the equalizing sequence of sheaves and five lemma, the above statement holds for global sections when $\mathscr{F}$ is quasicoherent. More precisely, when $\mathscr{F}$ is quasicoherent, for any $s\in \Gamma(X,\mathscr{O}_X)$ the canonical morphism $$ \Gamma(X,\mathscr{F})_s\to \Gamma(X_s,\mathscr{F}) $$ is an isomorphism.
I am interested in in the converse. I believe it is not true in general, so I am seeking a counter-example:
Give a qcqs scheme $X$ and an $\mathscr{O}_X$-module $\mathscr{F}$ such that for any $s\in \Gamma(X,\mathscr{O}_X)$ the canonical morphism $$ \Gamma(X,\mathscr{F})_s\to \Gamma(X_s,\mathscr{F}) $$ is an isomorphism but $\mathscr{F}$ is not quasicoherent. Further, can we find an example where $X$ is affine, or is the statement true for affine $X$?
For a counterexample, consider $X = \mathbb P^1_k$, and let $\mathscr F$ be any sheaf of $\mathcal O_X$-modules. Note that $\Gamma(X,\mathcal O_X) = k$, and every nonzero element in $k$ is already invertible on all of $X$. Thus, $X_s = X$ for all $s \in \Gamma(X,\mathcal O_X)\setminus\{0\}$, so the condition on $\mathscr F$ is vacuous.
(To find a sheaf of $\mathcal O_X$-modules which is not quasi-coherent, take for example $j_! \mathcal O_U$, where $U = D(x_0) \cong \mathbb A^1_k$ is a standard open, and $j \colon U \to X$ the inclusion.)