My first quess would be that the AC adaptor is starting to fail. It appears that when just charging the battery the output is high enought to charge it. However when you try to power the laptop the current requirement is higher and the output of the ac adaptor is dropping off.
1) The easyist with a know good adaptor. Any 19V/4 Amp AC Adaptor with the same plug should work.
2) the adaptor can be checked, but requires the use of a voltmeter (DVM).
Cost is about 15->20 Dollars (@ walmart) and they are easy to use (Just google how to use a voltmeter.). They are VERY handy around the House and for Autos - Not just for "computers"
steps.
This first step requires caution. Obtain two straight pins. There are two wires that go to the plug, you want to stick one pin in one of the wires and about a 1/4 in away stick the other pin. The Caution - do NOT stick pin in so that it shorts out the two wires. Reason for this is that you want to measure the voltage under load, which means plugging the connector into the laptop.
Next step is to measure the unloaded voltage by puting one meter lead to the barrel and the other lead to the inside hole. Plug adaptor into AC. Output DC should be above 19 Volts. Note if you read a neg voltage, reverse the meter leads. If you read zero volt, one of the two pins is shorting both wires unplug and redo.
Next, plug adaptor into laptop (laptop OFF). remeasure the voltage, should see a slight drop. Then power on the laptop, the voltage will probably drop slightly more than with the when just charging the battery. If voltage drops significantly - Ditch the adaptor and buy a new one.
AS I said, for most, it easier to just try a known good adaptor.