If $A$ is a set of reals such that $A \cap L[x] \in L[x]$ for each real $x$, is there a real $z$ such that $A \cap L[x] \in OD_z^{L[x]}$ for a cone of $x$?

This can be proved under the Axiom of Determinacy using a game of roughly the same complexity as $A$.

Just assuming boldface ${\bf \Delta}^1_2$ determinacy, if $A\cap L[x] \in OD^{L[x]}_z$ for a cone of $x$ then it can be shown that $A$ itself is $OD_z$. So perhaps one can find a counterexample by finding a homogeneous forcing that does not add reals but adds a set $A$ of reals with $A \cap L[x] \in L[x]$ for each real $x$.

(The assumption of ${\bf \Delta}^1_2$ determinacy is necessary here because starting with $V=L$ it is possible to force a new set $A$ with $A\cap L[x] \in OD^{L[x]}$ for each real $x$ by a homogeneous forcing that does not add reals.)