Carrier frequency offset (CFO) arises from the intrinsic mismatch between the operating frequencies of the transmitter and the receiver, as well as their relative speeds (i.e., Doppler effect). Despite advances in CFO estimation techniques, estimation errors are still present. Residual CFO creates time-varying phase error. Modern wireless systems, including WLANs, 5G cellular systems, and satellite communications, use high-order modulation schemes, which are characterized by dense constellation maps. Accounting for the phase error is critical for the demodulation performance of such schemes. In this paper, we analyze the post-estimation probability distribution of residual CFO and use it to develop a CFO-aware demodulation approach for a set of modulation schemes (e.g., QAM and APSK). For a given distribution of the residual CFO, symbols with larger amplitudes are less densely distributed on the constellation map. We explore one important application of our adaptive demodulation approach in the context of PHY-layer security, and more specifically modulation obfuscation (MO) mechanisms. In such mechanisms, the transmitter attempts to hide the modulation order of a frame's payload from eavesdroppers, which could otherwise exploit such information to breach user privacy or launch selective attacks. We go further and complement our CFO-aware demodulation scheme by optimizing the design of a low-complexity MO technique with respect to phase errors. Our results show that when combined, our CFO-aware demodulation and optimized MO techniques achieve up to 5 dB gain over conventional demodulation schemes that are not obfuscated and are oblivious to residual CFO.