This article considers the localization of a stationary target using time-delay and Doppler shift measurements obtained by moving monostatic sensors that takesthe sensor motion during the localization period into consideration. Starting from the model equation of a previous work, we propose a new formulation of optimization through constrained weighted least-squares to solve for the target location by developing two new localization solutions, one in closed form and the other by semi-definite relaxation (SDR). The new formulation enables the reduction of the minimum number of auxiliary variables in the closed-form solution and allows additional constraints in the SDR solution, resulting in fewer sensors needed in the former and less relaxation error and higher noise tolerance in the latter. Simulations validate the better performance of the proposed solutions than the previous ones.