Violations of the Born-Oppenheimer approximation (BOA) and the consequent nonadiabatic dynamics have long been an object of intense study. Recently, such dynamics have been induced via strong coupling of the molecule to a high-amplitude (spatially confined) mode of the electromagnetic field in optical cavities. However, the effects of a cavity on a pre-existing avoided crossing or conical intersection are relatively unexplored. The dynamics of molecules dressed by cavity modes are usually calculated by invoking the rotating wave approximation (RWA), which greatly simplifies the calculation but breaks down when the cavity mode frequency is higher than the relevant material frequencies. We develop a protocol for computing curve crossing dynamics in an optical cavity by exploiting a recently-developed method of solving the quantum Rabi model without invoking the RWA. The method is demonstrated for sodium iodide.