Pressure monitoring has an important significance in fields such as petrochemical, energy, and power engineering. Compared to electronic pressure sensors, fiber optic grating pressure sensors have the advantages of small size, simple structure, and anti-electromagnetic interference. This paper proposes a polymer encapsulated fiber Bragg grating (FBG) dynamic pressure sensor and uses a semiconductor optical amplifier (SOA) - fiber ring laser (FRL) and an array waveguide grating (AWG) demodulator to form a fiber Bragg grating dynamic pressure sensing demodulation system. The experimental results show that the laser output of the SOA-FRL system is stable; in the fiber Bragg grating pressure test, the pressure sensitivity of the dynamic pressure sensor based on SOA-FRL can reach -33.17 pm/MPa at two pressure environments of 1~ 20 MPa and 0.1~0.8 MPa. The temperature dependence test shows that the sensitivity of the dynamic pressure sensor is 8.14 pm/℃ in the temperature range of 23 ~ 60℃. The temperature sensitivity of the dynamic pressure sensor is slightly lower than that of the reference FBG sensor (9.79 pm/℃). In addition, the effects of different sensitizing materials on the sensitivity of the pressure sensor are compared. The results show that the sensitivity of polymer materials is higher than that of metal materials. Because the proposed dynamic pressure sensor system based on fiber ring laser and array waveguide grating demodulator has the characteristics of high sensitivity, simple structure, and the demodulation of dynamic signals, it has a promising application prospect in oil exploitation, transportation, structural monitoring and other fields.