Integrated sensing and communication (ISAC) has been envisioned as a solution to realize the sensing capability required for emerging applications in wireless networks, while efficiently utilizing the available spectral, hardware and energy resources. A commonly studied scenario in the ISAC literature involves a transceiver (e.g., a base station) serving a communication user, while simultaneously sensing targets from the reflected echo signals, by acting as a mono-static radar. However, as signal transmission durations are typically much longer than the radar echo round-trip times, the radar returns are drowned by the strong self interference (SI) from the transmitter - a phenomenon termed the echo-miss problem. A promising approach involves the ISAC transceiver to be full-duplex (FD), and in this paper we jointly design the transmit and receive beamformers at the transceiver, transmit precoder at the uplink user, and receive combiner at the downlink user to simultaneously (a) maximize the uplink and downlink rate, (b) maximize the transmit and receive radar beampattern power at the target, and (c) suppress the residual SI. Numerical results illustrate that the proposed design can effectively achieve up to 60 dB digital-domain SI cancellation, a higher average sum-rate, and more accurate radar parameter estimation compared with previous ISAC FD studies.