We have studied the effects of changing the barrier design of GaAs-AlxGa1-xAs quantum wells on the electroabsorption, exciton saturation, and carrier sweep-out times. Five samples have been studied with x values ranging from 0.2 to 0.4, and barrier thicknesses from 35 to 95 angstrom. Within this range, we find that the electroabsorption is not very sensitive to the barrier thickness, but that the ionization field of the excitons approximately doubles for an increase of x from 0.2 to 0.4. The samples with high, thick barriers have lower internal quantum efficiencies than those with low, thin barriers. We find that the exciton saturation intensity increases with increasing applied field, and decreasing barrier thickness or height. These findings are adequately explained by the field and barrier dependence of the thermionic emission and tunneling sweep-out rates. Time-resolved electroabsorption measurements confirm our understanding of the variation in sweep-out rates between samples, and indicate that the escape mechanism at low field is probably a thermally-assisted tunneling process.