Is my laptop broken or my power adapter broken?

It has nothing to do with your laptop. It's the power adapter. The original one is broken, I think. and I'm afraid the second one is broken too. I don't know where did you bought this adapter and how much you got it, but I'm sure it's poor quality.
I had adapter problem before, not the same problem, it just stoped working. I got a replacement from RAVPower and it works fine. I think you should have a try. And make sure you get the right adapter, check the output voltage and current, and also the connector size. Wish you good luck!

1) Remove battery, and try AC Adaptor. If it powers the laptop without problem, then I would expect the problem to be the battery, or the charger. If still a problem then Laptop, or charger.

2) while it can be done, it is difficault to determine which is which other than trying a KNOWN good charger.

3) To test you need a Digital volt meter (About $17 @ walmart), and the electrical ability to either make an adaptor cable, or ability to splice into the cable coming from the AC adaptor.
.. To measure voltage, you simply go across the two wires for 19 VDC.
.. To measure the current you have to cut a wire and put the meter in series.

If the Current is above the rated value - it's the laptop.
If Current is Below the rated value and the Voltage is low, it's the AC adaptor.

DEAD WRONG!!!!!!!!!

You don't need to splice anything. Google, or youtube much?

daship
DON'T need to Youtube or Google. This is simple Electronics 101!!!!
There is only two ways to measure current:
1) Place the meter (DVM) leads in series with the load. This can be done my making an adaptor cable with either the Positive or the negative wire cut and one lead goes to one end of the wire and the other lead goes to the other end thus placing the meter in series.
2) Use a current probe. However the current probe must go around only one of the wires, not both. Which means use an adaptor cable with access to the individual wires. They do make clamp on ampmeters, Or in my case I use a current probe that plugs into my O'scope so I can measure Inrush current.

To measure voltage UNDER LOAD, Just where would you stick the leads without disassembling the laptop and measure at the Jack.

NOTE measuring the plug with NO load is almost useless - Only checks for a DEAD Brick and not one that loads down under load..

Alternate method: Set up an external load to plug the brick into. Use a 6 Ohm 60 WATT resistor. Measure the Voltage across the load resistor approx 19 Volts and the e/6 ohms equals the current. In this case only interest in seeing if the brick can suppy 3.2 Amps without significant voltage drop. But then 60 Watt resistor is hard to find.