Self-interference (SI) is a known but critical issue at the receiver side of a monostatic radar in joint communications and sensing (JCAS) systems, and this is particularly true for high spectral-efficient waveforms. Full-duplex transceivers are usually assumed in the literature, but it is arguable if they can be taken for granted in many devices. We propose a different approach, by eliminating the SI using zero-padded (ZP) orthogonal frequency-division multiplexing (OFDM) and single-carrier frequency-division multiple access (SC-FDMA), instead of the more widespread cyclic-prefix (CP)-OFDM. ZP-sequences do not need full-duplex for the monostatic radar operation, as there is no SI between the transmit and receive antennas during the guard interval (GI), which can be used for radar detection. We derive the required radar receiver processing for ZP-sequences and CP-OFDM in time and frequency domains, respectively, to show that when the SI in JCAS is high, ZP-sequences can be beneficial, depending on the target range. Furthermore, we prove analytically that the low peak-to-average power ratio (PAPR) of ZP-SC-FDMA, when compared to ZP-OFDM, is beneficial in time-domain radar processing. This is demonstrated also by means of numerical simulation with ROC curves, for all the candidate waveforms in both coherent and incoherent receiver processing.