This paper focuses on showing time-message trade-offs in distributed algorithms for fundamental problems such as leader election, broadcast, spanning tree (ST), minimum spanning tree (MST), minimum cut, and many graph verification problems. We consider the synchronous CONGEST distributed computing model and assume that each node has initial knowledge of itself and the identifiers of its neighbors - the so-called KT1 model - a well-studied model that also naturally arises in many applications. Recently, it has been established that one can obtain (almost) singularly optimal algorithms, i.e., algorithms that have simultaneously optimal time and message complexity (up to polylogarithmic factors), for many fundamental problems in the standard KT0 model (where nodes have only local knowledge of themselves and not their neighbors). The situation is less clear in the KT1 model. In this paper, we present several new distributed algorithms in the KT1 model that trade off between time and message complexity. Our distributed algorithms are based on a uniform approach which involves constructing a sparsified spanning subgraph of the original graph - called a danner - that trades off the number of edges with the diameter of the sparsifier. In particular, a key ingredient of our approach is a distributed randomized algorithm that, given a graph G and any delta in [0, 1], with high probability constructs a danner that has diameter Otilde(D + n^(1 - delta)) and Otilde(min{m, n^(1 + delta)}) edges in Otilde(n^(1 - delta)) rounds while using Otilde(min{m, n^(1 + delta)}) messages, where n, m, and D are the number of nodes, edges, and the diameter of G, respectively. Using our danner construction, we present a family of distributed randomized algorithms for various fundamental problems that exhibit a trade-off between message and time complexity and that improve over previous results.