In this paper, we investigate the coexistence problem between Wi-Fi and a pre-standard form of LTE over unlicensed bands, namely, LTE-Unlicensed (LTE-U). We address two coexistence problems. First, the different access mechanisms for Wi-Fi and LTE-U can lead to an increase in the collision rate and higher latency for both systems. We propose a modified Wi-Fi operation mode, whereby Wi-Fi stations (STAs) carry out simultaneous spectrum sensing and transmission to reduce the time required for collision detection. Specifically, we propose and analyze a full-duplex (FD) based detection framework that can differentiate between Wi-Fi and LTE-U signals while taking into account residual self- interference. Second, the ability to differentiate between Wi-Fi and LTE-U signals motivates the idea of adapting the clear channel assessment (CCA) threshold according to the type of the detected signal. Inspired by upcoming Wi-Fi standards (e.g., IEEE 802.11ax), we propose a CCA threshold adaptation scheme and study via simulations its optimal setting so as to maximize the spatial reuse while maintaining fairness between LTE-U and Wi-Fi systems.