A linear-correction method is developed for source position and velocity estimation using time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements. The proposed technique first obtains an initial source location estimate using the first-step processing of an existing algebraic algorithm. It then refines the initial localization result by estimating via weighted least-squares (WLS) optimization and subtracting out its estimation error. The new solution is shown to be able to achieve the Cramer-Rao lower bound (CRLB) accuracy and it has better accuracy over several benchmark methods at relatively high noise levels.