Failure to recognize the AC adapter is not connected to the battery.  One of three parts is faulty: The adapter itself, The DC input jack,  The system board. Start with the AC adapter as you propose.  If a new, OEM adapter isn't recognized, replace the DC jack. [URL]   If that does not solve the problem, the charge circuit on the system board is faulty.  Replace the system board.  
I have an Inspiron 5570 that has issue with its ac adapter which shows Bios error Unknown Ac Adapter .... ..  slows  / underclock the CPU(current CPU speed around 0.40ghz to .80ghz) This also caused my battery to completely drained and pretty much a dead battery.I did some testing and found out that my center pin isn't outputting the enough voltage. Would buying a new Original Ac Adapter fixes bios error without buying a new battery? or would it need both ac adapter and battery just so it would not show a bios error and underclocks the CPU when used.I'm okay with it not having battery since it doesn't leave my table anyway since I have a separate laptop for outside use.  
dell inspiron 5570 ac adapter