Affine frequency division multiplexing (AFDM) is an emerging multicarrier waveform that offers a potential solution for achieving reliable communications over time-varying channels. This paper proposes two maximum-likelihood (ML) estimators of symbol time offset and carrier frequency offset for AFDM systems. One is called joint ML estimator, which evaluates the arrival time and carrier frequency offset by comparing the correlations of samples. Moreover, we propose the other so-called stepwise ML estimator to reduce the complexity. Both proposed estimators exploit the redundant information contained within the chirp-periodic prefix inherent in AFDM symbols, thus dispensing with any additional pilots. To further mitigate the intercarrier interference resulting from the residual frequency offset, we design a mirror-mapping-based scheme for AFDM systems. Numerical results verify the effectiveness of the proposed time and carrier frequency offset estimation criteria and the mirror-mapping-based modulation for AFDM systems.