To deeply understand the breakup mechanisms and characteristics of spray atomization of transverse fuel jet at the outlet of aero engine main combustor swirler, an adaptive mesh refinement method was employed to conduct primary atomization simulations under various conditions. By comparing the breakup structures of spray columns using different grid refinement levels, the independence of the adaptive refined strategy was verified. Furthermore, simulations of spray breakup and atomization were conducted under various inlet conditions, pressure drops, fuel flow rates, and fuel temperatures. These simulations revealed the structural characteristics of primary atomization and the statistical distributions of fuel droplets at the completion of primary atomization. The grid independence comparison showed that the primary atomization flow field structures obtained from simulations with 2, 3, and 4 levels of mesh refinement were consistent, indicating convergence of the adaptive refined refinement. Under four operating conditions—ground start, low speed, cruise, and design point—the breakup modes of the fuel spray columns exhibited distinct differences. At ground start condition, the Weber number is low, resulting the breakup morphology is columnar breakup, while at high Weber numbers, shear-induced breakup dominated. As the Weber number increased, the penetration depth of the liquid column extended, and the droplets distribution exhibited non-symmetric characteristics. With increasing inlet pressure and temperature, the characteristic diameters of primary atomization particles grew larger. When the swirler air pressure drop increased, the penetration depth of the spray column showed no significant change, while the characteristic diameters of primary atomization particles consistently increased. An increase in fuel flow rate resulted in the spray morphology being influenced by a combination of the air-to-liquid momentum ratio and Weber number. The penetration depth of the spray column increased, breakup occurred more rapidly, and the characteristic diameters of primary atomization droplets decreased. An increase in fuel temperature had no significant effect on the breakup morphology of the fuel under the present setting condition, though the characteristic diameters of primary atomization particles did increase with rising temperature.