A scheme for remote weak-signal sensors is proposed,in which a coupled-resonator optical waveguide(CROW),as a transmitter,couples to a hybrid optomechanical cavity and an observing cavity at its two ends.Non-Markovian theory is employed to study the weak-force sensor by treating the CROW as a non-Markovian reservoir of cavity fields.The dissipationless bound states in the non-Markovian regime are conducive to remotely transmitting a signal in the CROW.Our results show that a sensor with ultrahigh sensitivity can be achieved with the assistance of bound states under certain parameter regimes.