Does the system overheat? Were there any bluescreens / system crashes? Does this happen with or without the adapter connected?
You could run the LCD built in self test - Turn off the system - press and hold the "D" key and Power button to turn on the system - it should initiate the LCD test - see this link for more info - http://dell.to/1SkXNrG
Since the issue is the same on the external monitor as well, I suppose the video card is failing and hence you see the distorted video. And the motherboard needs to be replaced to resolve the issue.
Restart the system and press f12 key on startup - choose diagnostics. Let it run the diags and if you encounter any errors - make a note of the complete error and report the same to us.
You could write me a private message with the service tag, your name and contact details and we can help book a paid service and replace the parts, if the required parts are in stock, or for faster communication, you could contact us via Twitter - @dellcarespro
Personally, I would suggest you to invest your money on a new machine than to repair this system, as it would not support newer technology.