A new circularly pulse-shaped (CPS) precoding orthogonal frequency division multiplexing (OFDM) waveform, or CPS-OFDM for short, is proposed in this paper. CPS-OFDM, characterized by user-specific precoder flexibility, possesses the advantages of both low out-of-subband emission (OSBE) and low peak-to-average power ratio (PAPR), which are two major desired physical layer signal properties for various scenarios in 5G New Radio (NR), including fragmented spectrum access, new types of user equipments (UEs), and communications at high carrier frequencies. As opposed to most of existing waveform candidates using windowing or filtering techniques, CPS-OFDM prevents block extension that causes extra inter-block interference (IBI) and envelope fluctuation unfriendly to signal reception and power amplifier (PA) efficiency, respectively. An optimization problem of the prototype shaping vector built in the CPS precoder is formulated to minimize the variance of instantaneous power (VIP) with controllable OSBE power (OSBEP) and noise enhancement penalty (NEP). In order to solve the optimization problem involving a quartic objective function, the majorization-minimization (MM) algorithmic framework is exploited. By proving the convexity of the proposed problem, the globally optimal solution invariant of coming data is guaranteed to be attained via numbers of iterations. Simulation results demonstrate the advantages of the proposed scheme in terms of detection reliability and spectral efficiency for practical 5G cases such as asynchronous transmissions and mixed numerologies.