When the LED on the adapter goes out when it's connected to the system, there's a short to ground somewhere.  The shop was probably correct in replacing the adapter first -- without a known working adapter, there's no way to proceed with troubleshooting. If the adapter did not solve the problem, the two next places to look are at the DC jack (input) and at all the USB ports to be sure they're undamaged.  If any of them appears damaged, that would be next -- replace the jack.  If the USB ports is damaged, that generally leads to the need to replace the system board itself.  
So yesterday everything was working fine. I play a few games of league, chatting with some friends on discord, then I went afk to cook dinner when I came back I found my computer completely shut down. I try to turn in on and get nothing. Then I noticed the chargers led light was turned off.So I unplug it from the computer and the surge protector. After waiting a bit I plug it back into the surge protector and the led light turned back on but the moment I plug into the laptop it turns off. I took my lap top to a computer repair place. he said my charger is only outputting 4 watts when it should be 125w but he tried a 180 w and it still did nothing and said it was the power supply most likely and told me a dell one would be 125 so I went with that. now my issue (while I’m no computer repair guy) is out of all the different stuff relating to this same exact problem almost all of them seem to be mainboard issues. I wish I would have did a little more googling because now I’m worried that I needlessly spent 125 on an item I may not have needed and will have to spend even more on a mainboard when that part comes in and it doesn’t fix the issue. could a faulty power supply in an Alienware 17 R4 that was purchased back in 2017 really be the cause or did I just waste 125 dollars
17 R4, won’t turn on