Dissociative electron attachment (DEA) plays a key role in radiation damage of biomolecules under high-energy radiation conditions. The initial step in DEA is often rationalized in terms of resonant electron capture into one of the metastable valence states of a molecule followed by its fragmentation. Our combined theoretical and experimental investigations indicate that the manifold of states responsible for electron capture in the DEA process can be dominated by core-excited (shake-up) dipole-supported resonances. Specifically, we present the results of experimental and computational studies of the gas-phase DEA to three prototypical peptide molecules, formamide, N-methylformamide (NMF), and N,N-dimethyl-formamide (DMF). In contrast to the case of electron capture by positively charged peptides in which amide bond rupture is rare compared to N─C_{α} bond cleavage, fragmentation of the amide bond was observed in each of these three molecules. The ion yield curves for ions resulting from this amide bond cleavage, such as NH_{2}^{-} for formamide, NHCH_{3}^{-} for NMF, and N(CH_{3})_{2}^{-} for DMF, showed a double-peak structure in the region between 5 and 8 eV. The peaks are assigned to Feshbach resonances including core-excited dipole-supported resonances populated upon electron attachment based on high-level electronic structure calculations. Moreover, the lower energy peak is attributed to formation of the core-excited resonance that correlates with the triplet state of the neutral molecule. The latter process highlights the role of optically spin-forbidden transitions promoted by electron impact in the DEA process.