Perfect Doppler compensation and synchronization is nontrivial due to multi-path Doppler effects and Einstein's theory of relativity in the space-air-ground-integrated networks (SAGINs). Hence, by considering the residual Doppler and the synchronization delay, this paper investigates the bit-error-rate (BER) performance attained under time-varying correlated Shadowed-Rician SAGIN channels. First, a practical SAGIN model is harnessed, encompassing correlated Shadowed-Rician channels, the Snell's law-based path loss, atmospheric absorption, the line-of-sight Doppler compensation, elliptical satellite orbits, and Einstein's theory of relativity. Then, a specific correlation coefficient between the pilot and data symbols is derived in the context of correlated Shadowed-Rician channels. By exploiting this correlation coefficient, the channel distribution is mimicked by a bi-variate Gamma distribution. Then, a closed-form BER formula is derived under employing least-square channel estimation and equalization for 16-QAM. Our analytical results indicate for a 300-km-altitude LEO that 1) the period of realistic elliptical orbits is around 0.8 seconds longer than that of the idealized circular orbits; and 2) the relativistic delay is lower than 1 microsecond over a full LEO pass (from rise to set). Our numerical results for the L bands quantify the effects of: 1) the residual Doppler; 2) atmospheric shadowing; 3) synchronization errors; and 4) pilot overhead.