First experimental results from a room-temperature tabletop phase-sensitive axion haloscope experiment are presented. The technique exploits the axion-photon coupling between two photonic resonator oscillators excited in a single cavity, allowing low-mass axions to be upconverted to microwave frequencies, acting as a source of frequency modulation on the microwave carriers. This new pathway to axion detection has certain advantages over the traditional haloscope method, particularly in targeting axions below 1 μeV (240 MHz) in energy. At the heart of the dual-mode oscillator, a tunable cylindrical microwave cavity supports a pair of orthogonally polarized modes (TM_{0,2,0} and TE_{0,1,1}), which, in general, enables simultaneous sensitivity to axions with masses corresponding to the sum and difference of the microwave frequencies. However, in the reported experiment, the configuration was such that the sum frequency sensitivity was suppressed, while the difference frequency sensitivity was enhanced. The results place axion exclusion limits between 7.44-19.38 neV, excluding a minimal coupling strength above 5×10^{-7} 1/GeV, after a measurement period of two and a half hours. We show that a state-of-the-art frequency-stabilized cryogenic implementation of this technique, ambitious but realizable, may achieve the best limits in a vast range of axion space.