Dependent rounding is a useful technique for optimization problems with hard budget constraints. This framework naturally leads to negative correlation properties. However, what if an application naturally calls for dependent rounding on the one hand, and desires positive correlation on the other? More generally, we develop algorithms that guarantee the known properties of dependent rounding, but also have nearly best-possible behavior - near-independence, which generalizes positive correlation - on "small" subsets of the variables. The recent breakthrough of Li & Svensson for the classical k-median problem has to handle positive correlation in certain dependent-rounding settings, and does so implicitly. We improve upon Li-Svensson's approximation ratio for k-median from 2.732+? to 2.611+? by developing an algorithm that improves upon various aspects of their work. Our dependent-rounding approach helps us improve the dependence of the runtime on the parameter ? from Li-Svensson's N^O(1/?^2) to N^O((1/?)log(1/?)).