Extremely variable quasars can also show strong changes in broad-line emission strength and are known as changing-look quasars (CLQs). To study the CLQ transition mechanism, we present a pilot sample of CLQs with X-ray observations in both the bright and faint states. From a sample of quasars with bright-state archival SDSS spectra and (Chandra or XMM-Newton) X-ray data, we identified five new CLQs via optical spectroscopic follow-up and then obtained new target-of-opportunity X-ray observations with Chandra. No strong absorption is detected in either the bright- or the faint-state X-ray spectra. The intrinsic X-ray flux generally changes along with the optical variability, and the X-ray power-law slope becomes harder in the faint state. Large-amplitude mid-infrared variability is detected in all five CLQs, and it echoes the variability in the optical with a time lag expected from the light-crossing time of the dusty torus for CLQs with robust lag measurements. The changing-obscuration model is not consistent with the observed X-ray spectra and spectral energy distribution changes seen in these CLQs. It is highly likely that the observed changes are due to the changing accretion rate of the supermassive black hole, so the multiwavelength emission varies accordingly, with promising analogies to the accretion states of X-ray binaries.