This paper introduces a novel channel and carrierafrequencyaoffset (CFO) estimation scheme for orthogonalafrequency-divisionamultiplexing (OFDM) transmission over time-varying channels. The proposed design is highly flexible and compatible with orthogonalamultipleaaccess (OMA) and non-orthogonalamultipleaaccess (NOMA) transmission. The new estimator performs the channel and CFO estimation jointly and blindly, highlighting the efficiency of the proposed estimator. The blind CFO and channel estimation are realized by developing a new OFDM frame layout, where phaseashiftakeying (PSK) and amplitudeashiftakeying (ASK) symbols are used to modulate specific subcarriers over successive OFDM frames. An arbitrary modulation scheme can be used to modulate all other subcarriers. The new frame layout enables the use of amplitudeacoherentadetection (ACD) and Viterbi-and-Viterbi (VAV), which are used to perform blind channel and CFO estimation and compensation. A closed-form analytical formula is derived for the exact symbolaerrorarate (SER) of the ASK symbols, which is then used to derive a precise formula for the meanasquaredaerror (MSE) for the OMA case. For NOMA, the same approach can be used and the analysis is generally similar, hence, Monte Carlo simulation is invoked to evaluate the performance for the NOMA case. The obtained results show that the new frame layout can improve the spectral efficiency while enabling accurate channel and CFO estimation for both OMA and NOMA with computational complexity analogous to pilot-assisted schemes. The improved spectral efficiency is due to the replacement of pilot symbols with data-bearing symbols. The performance of the system is evaluated in terms of MSE and SER for a wide range of operating scenarios, and the results confirm the robustness and reliability of the proposed scheme for both OMA and NOMA.