In inertial confined fusion (ICF) experiments, the temporal evolution of X-ray spectrum can provide important diagnostic information such as electron temperature and density on laser-plasma interaction. Accurate diagnostic requires a wide range of X-ray spectrum from several hundred eV to kilo eV to be measured with high temporal resolution. For traditional single grating spectrometer coupled with streak cameras, the limited recording length of streak cameras severely restricts measured X-ray spectral range in one laser shot. Here we design a shifted dual transmission grating (SDTG) spectrometer for laser-produced plasma X-ray diagnostics in ICF experiments which can provide wide-range X-ray spectrum measurement from 100 eV to 5 keV with high temporal and spectral resolution. This SDTG spectrometer comprises two X-ray gratings: one with high line density and the other with low line density. The high line density grating is used to measure X-ray spectrum from 1000 eV to 5000 eV and the low line density grating measures X-ray spectrum from 100 eV to 1000 eV respectively. These two kinds of X-ray gratings are arranged in a plane with their centers shifted by a certain distance. A shifted double slit component is designed according to the spatial positions of the two gratings and set in front of the photocathode in the streak camera to ensure that two sets of X-ray spectra by two shifted gratings are projected on the photocathode without overlapping. This novel SDTG-based X-ray spectrometer can take the most of recording panel space, offering a path to realize a high resolution and broad spectral ranges in diagnosing soft X-rays. In this paper, the design method and the technical data of the SDTG-based X-ray spectrometer are given. The SDTG-based X-ray spectrometer is integrated, debugged and used to measure X-ray pulse at SG-III prototype facility located in Laser Fusion Research Center of Chinese Academy of Engineering Physics. The time integral results are captured by the SDTG spectrometer in the ICF fluid RT experiments and time-resolved spectra are recorded in indirect drive implosion experiment. Experimental results show the SDTG-based X-ray spectrometer can capture X-ray spectrum ranging from 0.1 keV to 5 keV, with a spectral resolution of 0.04 nm and a temporal resolution of better than 30 ps. By fully utilizing limited recording length, the SDTG-based X-ray spectrometer can realize a wide range temporal X-ray spectrum measurement with enough spectral resolution and temporal resolution. This SDTG spectrometer is a good temporal X-ray diagnostic tool for ICF experiments and other high energy density physics experiments.