Ultrathin nanostructured membranes are widely pursued to apply into different processes ranging from air separation to seawater desalination. Here, freestanding carbon nanomembranes (CNMs) are employed to dehydrate vaporous alcohols at room temperature. The structure of the microporous material is addressed by measuring permeation rates of homologous n-alkanols. To examine the separation performance, we introduce a model heavy water/n-propanol azeotrope. While ordinary nanomembranes show moderate selectivity of around 300, complete rejection of organic molecules is achieved upon stacking two CNM layers. Furthermore, the mixture experiments with the stacks demonstrate a 10-fold slowdown in the transmembrane diffusion of water as compared to both the single-layer material and pure vapor. We discuss the observed effect as a "molecular jam" in the interlayer spacing, which effectively disrupts the collective flow of liquefied water. Our work sheds light on molecular transport under nanoconfinement.