We report on an optically-based technique that provides an efficient way to track static and dynamic strain by locking the frequency of a diode laser to a fiber Bragg-grating Fabry-Pérot cavity. For this purpose, a suitable optical frequency discriminator is generated exploiting the fiber natural birefringence and that resulting from the gratings inscription process. In our scheme, a polarization analyzer detects dispersive-shaped signals centered on the cavity resonances without need for additional optical elements in the resonator or any laser-modulation technique. This method prevents degradation of the resonator quality and maintains the configuration relatively simple, demonstrating static and dynamic mechanical sensing below the picostrain level.