A highly sensitive relative humidity (RH) sensor based on Fabry-Perot interferometers (FPI) is proposed and experimentally demonstrated. The sensor is fabricated by splicing a segment of hollow core Bragg fiber (HCBF) with single mode fiber (SMF) and functionalized with chitosan and ultraviolet optical adhesive (UVOA) composite at the end of HCBF to form a hygroscopic polymer film. The reflection beams from the splicing point and the two surfaces of the polymer film generate the Vernier effect in the reflection spectrum, which significantly improves the humidity sensitivity of the sensor. To demodulate the envelope based on the Vernier effect and realize multi-point sensing, a digital signal processing (DSP) algorithm is proposed to process the reflection spectrum. The performance of the DSP algorithm is theoretically analyzed and experimentally verified. The proposed sensor demonstrates a high sensitivity of 1.45 nm/% RH for RH ranging from 45% RH to 90% RH. The compact size, high sensitivity and multiplexing capability make this sensor a promising candidate for RH monitoring. Furthermore, the proposed DSP can potentially be applied to other sensors based on the Vernier effect to analyze and extract valuable information from the interference spectrum.