Before doing anything at all, make sure all drivers and software is up to date. if any updates are pending, be sure to finish them.
measure the voltage across the jack on your AC adapter. make sure its coming out at the required voltage, (as marked on the item) it is usually between 18-21 V depending on the lap top. if it doesnt read the proper voltage, then yes, replace the charger.
otherwise, you could be looking at a port failure, where the port isnt pulling the right current. open the device, and plug the adapter into the port, and measure the voltage at the point where the port connects to the board. If youre still getting the proper voltage there, then you are looking into board level issues, or software failure.
From here, back up any important data, and reformat your device. reinstall windows, and run your updates to see if this fixes your issues.
(after this i would say replace the battery, but you already have, and you are getting the same problem. so assuming you didnt order it 2nd hand off ebay or something, you can skip this)''
From here, its time to look at board level failures. if the board is failing, you would start by scanning for hardware damage. (blown filters and diodes, and the like.) if you find one, replace it. if you cant, you can always check continuity for components with your multimeter on the diode setting.
after all of this is said and done, if you dont find something, then a logic board replacement is in order.