We solve the time-dependent, compressible Poiseuille and extrudate-swell flows of a shear-thinning fluid that obeys the Carreau constitutive model, using finite elements in space and a fully-implicit scheme in time. Slip is assumed to occur along the die wall following a non-monotonic slip equation that relates the wall shear stress to the slip velocity and is based on experimental measurements with polyethylene melts. Thus, the resulting flow curve is also non-monotonic, and consists of two stable positive-slope branches and a linearly unstable negative-slope branch. The steady-state numerical results compare well with certain analytical solutions for Poiseuille flow. The time-dependent calculations at fixed volumetric flow rates demonstrate the existence of periodic solutions in the unstable regime, due to the combination of compressibility and slip. Self-sustained oscillations of the pressure-drop and of the mass-flow rate are obtained. In the extrudate region, high-frequency, small amplitude waves are generated on the free-surface, which also oscillates radially. The wavelength and the amplitude of the free-surface waves and the amplitude of the oscillations in the radial direction are reduced, as the Reynolds number is decreased and approaches the conditions of the experiments.