I would check the battery as they do die from age. Install this gem of an App [https://www.coconut-flavour.com/coconutbattery/|CoconutBattery] take a snapshot of the Apps main window while the charger is plugged in. and post it here so we can evaluate the batteries condition.
depending on the macOS you are running see if you can run the onboard diagnostics, Apple did mess up the system firmware with an update so it might not work, still give it a try. Restart the system and press the ***D*** key to enter. Let us know if you get any errors.
Reference: [https://support.apple.com/en-us/102603|Mac startup key combinations]