An interleaved, chirped electro-optic dual comb system is demonstrated for rapid, high dynamic range measurements of cavity optomechanical sensors. This approach allows for the cavity displacements to be interrogated at measurement times as fast as 10 µs over ranges far larger than can be achieved with alternative methods. While the performance of this novel, to the best of our knowledge, readout approach is evaluated with an optomechanical accelerometer, this method has a wide range of applications including temperature, pressure, and humidity sensing as well as acoustics and molecular spectroscopy.