Motivated by a problem in which a heteroclinic orbit represents a moving interface between ordered and disordered crystalline states, we consider a class of slow-fast Hamiltonian systems in which the slow manifold loses normal hyperbolicity due to a transcritical or pitchfork bifurcation as a slow variable changes. We show that under assumptions appropriate to the motivating problem, a singular heteroclinic solution gives rise to a true heteroclinic solution. In contrast to previous approaches to such problems, our approach uses blow-up of the bifurcation manifold, which allows geometric matching of inner and outer solutions.