Kim and Drake used generating functions to prove that the number of 2-distant noncrossing matchings, which are in bijection with little Schröder paths, is the same as the weight of Dyck paths in which downsteps from even height have weight 2. This work presents bijections from those Dyck paths to little Schröder paths, and from a similar set of Dyck paths to big Schröder paths. We show the effect of these bijections on the corresponding matchings, find generating functions for two new classes of lattice paths, and demonstrate a relationship with $231$-avoiding permutations.