Epsilon-near-zero (ENZ) nanophotonic devices with zero permittivity are known to exhibit adiabatic frequency translation via temporal refraction under extracavity excitation by intense light sources, which are however hard to integrate on-chip owing to a high demand for energy density. As this class of complementary-metaloxide-semiconductor-compatible materials is progressing toward on-chip photonic integration, a more versatile solution with less intensity requirements needs to be further explored. Here, for the first time, by leveraging the abundant frequency mode resources inside a resonant cavity, we experimentally demonstrate the realization of input-dependent dual-range frequency switching via a single intracavity ENZ element. By utilizing the linear and nonlinear effects induced by ENZ, the system can perform a small 279.73 GHz as well as a 13-octave-span larger (3.63-THz) mode-locked frequency shift at 196 and 192 THz, respectively, under a pulse energy 2 orders of magnitude lower than extracavity schemes with a conversion efficiency (in %frequency shift per unit energy density per unit material thickness) also 2 orders of magnitude higher. Additionally, we report for the first time the real-time observation of the intracavity ENZ frequency switching operation, proving that the mechanism differs from pure ENZ time refraction. We further discuss that by encoding the states of two intracavity components, the optical system can program eight types of different 1- and 2-operand logic functions, including four complex noncommutative ones. This work extends the understanding of ENZ photonics beyond extracavity scenarios. The proposed solution could be extended to photonic integration with a potential for novel optical logic gates and photonic computing designs as an efficient and simplified alternative to microelectronic counterparts.