Low half-wave-voltage electro-optic phase modulators can reduce system power consumption and simplify driving circuits, which are crucial in optical signal processing. Using X-cut thin-film lithium niobate photonic platforms and the principle of radio-frequency (RF) phase-delay modulation, we optimized the optical and electrical parameters of a cross-waveguide phase modulator with ground-signal-ground electrodes. Preliminary experiments were conducted to verify the modulator performance. At an in-phase modulation frequency of 17.5 GHz and a total modulation length of 10 mm, the RF voltage-length product was experimentally measured to be 7.5 V·cm, consistent with the design results. Increasing the total modulation length to 40 mm further reduced the half-wave voltage to 2.2 V. A passive waveguide transmission loss of approximately 0.4 dB/cm was measured from the co-integrated microring resonator. At a high RF driving power of 29 dBm, the operation of the cross-waveguide phase modulator in the high-frequency region of 30 GHz was verified. This research offers guidance for designing and analyzing on-chip electro-optic modulators with low half-wave voltages.