This paper studies the degrees of freedom of full-duplex multicell networks that share the spectrum among multiple cells in a non-orthogonal setting. In the considered network, we assume that full-duplex base stations with multiple transmit and receive antennas communicate with multiple single-antenna mobile users. By spectrum sharing among multiple cells and (simultaneously) enabling full-duplex radio, the network can utilize the spectrum more flexibly, but, at the same time, the network is subject to multiple sources of interference compared to a network with separately dedicated bands for distinct cells and uplink-downlink traffic. Consequently, to take advantage of the additional freedom in utilizing the spectrum, interference management is a crucial ingredient. In this work, we propose a novel strategy based on interference alignment which takes into account inter-cell interference and intracell interference caused by spectrum sharing and full-duplex to establish a general achievability result on the sum degrees of freedom of the considered network. Paired with an upper bound on the sum degrees of freedom, which is tight under certain conditions, we demonstrate how spectrum sharing and full-duplex can significantly improve the throughput over conventional cellular networks, especially for a network with large number of users and/or cells.