Grid-forming (GFM) voltage-source converters (VSCs) are required to provide two functionalities: ensuring synchronization stability, which is vital for their operation, and supplying GFM capability, which provides inertia power to support AC grid frequency. However, these functionalities posed a dilemma on the rate of convergence of synchronization (RoCoS) of the VSC. The synchronization stability can be enhanced with a higher RoCoS, while enhancing GFM capability requires a lower RoCoS to provide more inertia active power. To reconcile this dilemma, a control structure capable of seamless transitions between the high RoCoS and the low RoCoS is desirable. Previous adaptive fast/slow synchronization control methods depended on stiff DC-side sources and may not work in scenarios with limited DC-side power capacity. To address this limitation, the paper introduces an adaptable fast/slow synchronization control structure for a dual-port grid-forming (DGFM) VSC with an energy storage device (ESD). The proposed control structure allows adjustment of RoCoS through the Pdc/vdc droop coefficient of the ESD.