Does $\mathsf{D}\mathsf{C}$ imply ${\mathsf{A}\mathsf{C}}_{\omega }$, uniformly?

The Axiom of Dependent Choice $\mathsf{D}\mathsf{C}$ and the Axiom of Countable Choice ${\mathsf{A}\mathsf{C}}_{\omega }$ are two weak forms of the Axiom of Choice that can be stated for a specific set: $\mathsf{D}\mathsf{C}(X)$ asserts that any total binary relation on $X$ has an infinite chain, while ${\mathsf{A}\mathsf{C}}_{\omega }(X)$ asserts that any countable collection of nonempty subsets of $X$ has a choice function. It is well-known that $\mathsf{D}\mathsf{C}\Rightarrow {\mathsf{A}\mathsf{C}}_{\omega }$. We study for which sets and under which hypotheses $\mathsf{D}\mathsf{C}(X)\Rightarrow {\mathsf{A}\mathsf{C}}_{\omega }(X)$, and then we show it is consistent with $\mathsf{Z}\mathsf{F}$ that there is a set $A\subseteq \mathbb{R}$ for which $\mathsf{D}\mathsf{C}(A)$ holds, but ${\mathsf{A}\mathsf{C}}_{\omega }(A)$ fails.