The mechanisms underlying the collective motion of molecular motors in living cells are not yet fully understood. One such open puzzle is the observed pulses of backward moving myosin-X in the filopodia structure. Motivated by this phenomenon we introduce two generalizations of the 'Total Asymmetric Exclusion Process' (TASEP) that might be relevant to the formation of such pulses. The first is adding a nearest-neighbours attractive interaction between motors, while the second is adding an internal degree of freedom corresponding to a processive and immobile form of the motors. Switching between the two states occurs stochastically, without a conservation law. Both models show strong deviations from mean field behaviour and lack particle-hole symmetry. We use approximations borrowed from the research of vehicular traffic models to calculate the current and jam size distribution in a system with periodic boundary conditions and introduce a novel modification to one of these approximation schemes.