We describe analytically, and simulate numerically, traveling waves with oscillatory tails in a bistable, piecewise-linear reaction-diffusion-advection system of the FitzHugh-Nagumo type with linear cross-diffusion and cross-advection terms of opposite signs. We explore the dynamics of two wave types, namely, solitary pulses and their infinite sequences, i.e., periodic wave trains. The effects of cross diffusion and cross advection on wave profiles and speed of propagation are analyzed. For pulses, in the speed diagram splitting of a curve into several branches occurs, corresponding to different waves (wave branching). For wave trains, in the dispersion relation diagram there are oscillatory curves and the discontinuous curve of an isola with two branches. The corresponding wave trains have symmetric or asymmetric profiles. Numerical simulations show that for large values of the period there exist two wave trains, which come closer and closer together and are subject to fusion into one when the value of the period is decreasing. Other types of waves are also briefly discussed.