I also had similar issues and even had the dealer look at it. Everything was working correctly according to GM. Also noticed the loose fitting plug. Seemed too loose to maintain electrical connections. So I looked inside the receptacle and if you look closely you can see what looks like a quarter round piece of trim between the base and the wall. I figured my plug was going just above this, keeping it loose. Next time I towed, I had to push really hard to get the plug to seat inside that quarter round. It did! The plug was now tight, and never had the problem again. If you do this, just make sure the plug is aligned with the pins correctly and should be 95% of the way in, still loose fitting. That last 1/8 of inch is the push really hard part.
good luck.