Does DC imply AC_omega, uniformly?

The Axiom of Dependent Choice DC and the Axiom of Countable Choice ACω are two weak forms of the Axiom of Choice that can be stated for a specific set: DC(X) asserts that any total binary relation on X has an infinite chain, while ACω(X) asserts that any countable collection of nonempty subsets of X has a choice function. It is well-known that DC ⇒ ACω. We study for which sets and under which hypotheses DC(X) ⇒ ACω(X), and then we show it is consistent with ZF that there is a set A ⊆ R for which DC(A) holds, but ACω(A) fails.