This paper introduces a new adaptive control strategy for power-sharing in a hybrid AC/DC microgrid (HMG). The existing interlink converter (ILC) control methods exhibit limitations under heavy load and contingency conditions. To overcome this limitation, a bidirectional control strategy is proposed for the ILC, which effectively coordinates battery energy storage systems (BESSs) in both sub-grids while mitigating stress on the ILC and BESSs. The control strategy utilises the state of charge (SoC) and instantaneous power of the BESSs to stabilise frequency and DC voltage on the AC and DC sub-grids, respectively, enabling autonomous operation and plug-and-play capability. Stability analysis employing root locus and a stress level relaxation index for BESSs are presented. Performance evaluation based on the moving mean square error criterion validates the effectiveness of the control strategy. Simulation results using MATLAB/SIMULINK demonstrate the superior performance of the proposed strategy in regulating both voltage and frequency within the system over the existing control approaches. Furthermore, the proposed approach significantly reduces stress levels on the BESSs in different operation conditions.