The spread of disinformation (maliciously spread false information) in online social networks has become an important problem in today's society. Disinformation's spread is facilitated by the fact that individuals often accept false information based on cognitive biases which predispose them to believe information that they have heard repeatedly or that aligns with their beliefs. Moreover, disinformation often spreads in direct competition with a corresponding true information. To model these phenomena, we develop a model for two competing beliefs spreading on a social network, where individuals have an internal opinion that models their cognitive biases and modulates their likelihood of adopting one of the competing beliefs. By numerical simulations of an agent-based model and a mean-field description of the dynamics, we study how the long-term dynamics of the spreading process depends on the initial conditions for the number of spreaders and the initial opinion of the population. We find that the addition of cognitive biases enriches the transient dynamics of the spreading process, facilitating behavior such as the revival of a dying belief and the overturning of an initially widespread opinion. Finally, we study how external recruitment of spreaders can lead to the eventual dominance of one of the two beliefs.