A receiver architecture for cognitive sensing and navigation with orthogonal frequency division multiplexing (OFDM)-based systems is proposed. The proposed receiver enables exploiting all the transmitted periodic beacons of 5G new radio (NR) and Starlink low Earth orbit (LEO) signals to draw navigation observables. Reference signals (RSs) of modern OFDM-based systems, such as 5G NR, contain both always-on and on-demand components. These components can be unknown or known but subject to change. To leverage all transmitted signals for navigation purposes, the RS signals should be detected and tracked cognitively. Similar to conventional navigation receivers, the proposed architecture involves acquisition and tracking stages. However, both stages are supplemented by the unorthodox capability of estimating and updating the RS signals. The acquisition stage instructs the tracking stage by reporting performance metrics, which are used to adjust the tracking loop gains to update the RS accordingly. A chirp model is considered to capture the high dynamics of Doppler frequency in intensive Doppler scenarios, where the navigating vehicle is maneuvering or the transmitting source is not static. The effect of Doppler rate estimation error on frame length estimation is analyzed. Experimental results are presented demonstrating the performance of the proposed receiver by: (i) enabling an unmanned aerial vehicle (UAV) to detect and exploit terrestrial 5G NR cellular signals in a blind fashion for navigation purposes, achieving a two-dimensional (2D) root-mean squared error (RMSE) of 4.2 m over a total trajectory of 416 m; (ii) enabling a ground vehicle that traversed a trajectory of 1.79 km to cognitively sense an unknown gNB (blindly detect, track, and exploit transmitted always-on and on-demand signals), localizing it with a 2D error of 5.83 m; and (iii) tracking Starlink LEO OFDM signals, producing Doppler measurements, which were fused to localize a stationary receiver with a 2D error of 6.5 m, starting from an initial estimate 179 km away from the receiver's true position.