In Cognitive Radio (CR) networks, multiple secondary network users (SUs) attempt to communicate over wide potential spectrum without causing significant interference to the Primary Users (PUs). A spectrum sensing algorithm is a critical component of any sensing strategy. Performance of conventional spectrum detection methods is severely limited when the average SNR of the fading channel between the PU transmitter and the SU sensor is low. Cooperative sensing and advanced detection techniques only partially remedy this problem. A key limitation of conventional approaches is that the sensing threshold is determined from the miss detection rate averaged over the fading distribution. In this paper, the threshold is adapted to the instantaneous PU-to-SU Channel State Information (CSI) under the prescribed collision probability constraint, and a novel sensing strategy design is proposed for overlay CR network where the instantaneous false alarm probability is incorporated into the belief update and the reward computation. It is demonstrated that the proposed sensing approach improves SU confidence, randomizes sensing decisions, and significantly improves SU network throughput while satisfying the collision probability constraint to the PUs in the low average PU-to-SU SNR region. Moreover, the proposed adaptive sensing strategy is robust to mismatched and correlated fading CSI and improves significantly on conventional cooperative sensing techniques. Finally, joint adaptation to PU channel gain and SU link CSI is explored to further improve CR throughput and reduce SU collisions.