We report analytical and numerical investigation of pulse formation and spectral broadening in continuous-wave (CW)-pumped optical parametric oscillator in doubly-resonant oscillator (DRO) configuration operating at degeneracy with dispersion control. The work addresses fundamental limitations in conventional cw DRO that includes mode and cluster hopping, which hinder the generation of stable ultrashort pulses when explored for mode-locking. By implementing group-delay dispersion compensation, we demonstrate elimination of frequency instabilities and stable spectral and temporal characteristics under varying cavity and pump conditions. Using coupled-wave equations and the Split-Step Fourier Method, we analyzed the cold cavity dynamics and observed the incipient stage of pulse formation and enhanced spectral gain bandwidth. Further, we explored the mode-locked dispersion-compensated cw DRO and observed the phase locking of signal and idler modes, resulting in broadband multi-axial-mode output. We analyzed the carrier envelope offset phase, which is naturally stabilized in dispersion-compensated configurations, allowing passively-stable generation of ultrashort pulses without active stabilization. These findings establish a framework for realizing stable, phase-locked ultrashort pulse generation under cw pumping.