Which of the ADP5050/1/2/3/4 products are you looking at? If you're still deciding on which part to use, I suggest you also use the ADP505x Buck Regulator Design tool, set your design requirements and the tool will provide you with the proper components to use. Let me know if you need further help, or require other information.
Your operating currents across all rails are too high for the chip to handle simultaneously, and this will cause the die temperature to increase, and the part will go into thermal shutdown. Are these your max currents? I suggest that you check the tool for the ADP5054, to see more info on the estimated performance for your design. The die temp increases as well based on your ambient temperature.
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