(TMTTF)2AsF6 undergoes two phase transitions upon cooling from 300 K. At Tco=103 K a charge-ordering (CO) occurs, and at Tsp(B=9 T)=11 K the material undergoes a spin-Peierls (SP) transition. Within the intermediate, CO phase, the charge disproportionation ratio is found to be at least 3:1 from carbon-13 NMR 1/T1 measurements on spin-labeled samples. Above Tsp, up to about 3Tsp, 1/T1 is independent of temperature, indicative of low-dimensional magnetic correlations. With the application of about 0.15 GPa pressure, Tsp increases substantially, while Tco is rapidly suppressed, demonstrating that the two orders are competing. The experiments are compared to results obtained from calculations on the 1D extended Peierls-Hubbard model.