Q: WiFi drops (DNS) when plugging in charger

Is the magsafe getting overtly hot? You could use about this mac/systems/power to see what voltage it is putting out. Should not be over 12500MV.  https://support.apple.com/en-us/HT203207 You could also try a SMC reset.  https://support.apple.com/en-us/HT201295

Do a pram reset. https://support.apple.com/en-us/HT204063  When you say plugged in my connector. Do you mean ethernet cable or magsafe?  If you are running, wifi only, from your Macbook Air I do not see how that can shut down the internet completely thru wifi. Unless you have a problem with the wifi card. What happens if you turn off the wifi and run by ethernet cable?

I use ethernet when I am at my desk because the internet speed is better and more reliable. The reason I asked about trying with ethernet was to see if you get the same results. If not you could have a bad airport card. Since you have no problem in wifi when on battery only. It doesn't sound like a bad configuration on your computer.  You could go to the utilities folder, activity monitor, and then plug in the magsafe to see under power and  internet if you get a spike. You may still need to take it to the genius bar to be checked.