Several kinds of phase-shifted long-period gratings (LPGs) have been fabricated using a high-repetition-rate CO2 laser, where both the grating and the inserted phase shift are simultaneously produced in a commonly used single-mode fiber. Furthermore, using one of the fabricated phase-shifted LPGs, a temperature-insensitive sensor that allows for large axial strain measurement with a value up to 7600 mu epsilon, is proposed and first demonstrated. Measurement accuracy for the strain ranging from 340 to 7600 mu epsilon is estimated to be +/- 30 mu epsilon, while the temperature is changed in a range of 20 degrees C-140 degrees C.