Recently, searches were made for H2CCS and HCCSH in a variety of interstellar environments─all of them resulted in nondetections of these two species. Recent findings have indicated the importance of destruction pathways, e.g., with atomic hydrogen, in explaining the consistent nondetection of other species, such as the H2C3O family of isomers. We have thus performed ab initio calculations looking at reactions of H2CCS, HCCSH, and related species with atomic hydrogen. Our results show that H2CCS and HCCSH are both destroyed barrierlessly by atomic hydrogen, thus providing a plausible explanation for the nondetections. We further find that subsequent reactions with atomic hydrogen can barrierlessly lead to CH3CH2SH, which has been detected. Astrochemical simulations including these reactions result not only in reproducing the observed abundance of H2CCS in TMC-1 but also show that CH3CH2SH, produced via our H-addition pathways and subsequently trapped on grains, can desorb in warmer sources up to abundances that match previous observations of CH3CH2SH in Orion KL. These results, taken together, point to the importance of grain-surface H-atom addition reactions and highlight the chemical links between cold prestellar cores and their subsequent, warmer evolutionary stages.