There is a growing interest in the use of legacy terrestrial GPS signals to determine the precise positioning and timing onboard a lunar satellite. Unlike the prior works that utilize the meter-level accurate pseudoranges, we propose a precise positioning and timekeeping technique that leverages millimeter-level accurate carrier phase measurements (when integer ambiguities are correctly fixed). We design an extended Kalman filter framework that harnesses the intermittently available terrestrial GPS time-differenced carrier phase (TDCP) values and the gravitational accelerations predicted by the orbital filter. We implement an adaptive state noise compensation algorithm to estimate the process noise covariance that adapts to the challenging lunar environment with weak gravity and strong third-body perturbations. Additionally, we perform measurement residual analysis to discard TDCP measurements corrupted by cycle slips and increased measurement noise. We demonstrate Monte-Carlo simulations of a lunar satellite in the elliptical lunar frozen orbit (ELFO) and Quasi-Frozen Low Lunar Orbit (QFLLO), wherein we showcase higher positioning and timing accuracy as compared to the pseudorange-only navigation solution.