I would not say that full-frame is inherently "better" than APS-C for that kind of photography. It's 'different', and the 6D would have a decided edge over the 7D in really dark, low-light situations at high ISO levels, but I did not see any low-light, nightclub, cabaret, dinner theatre type stuff where the 6D would be a "better" tool. Use whatever camera you own or have, and practice with it.
You need a camera you are familiar with, and a lens set that you KNOW really well, so that the camera and the lens it has on it becomes totally transparent, totally "automatic" to shoot with. No futzing around, no dilly-dallying, zooming, no fiddle-farting around. One of the reasons world-famous shooter Henri Cartier-Bresson used a Leica and 50mm lens for over 98% of his pictures was, I think, because he knew EXACTLY what the 50mm lens would "see" when he shot. That is one of the things a prime lens does; after a while, the user KNOWS,beforehand, what the lens will see, in a wide range of scenarios, even before he brings the camera up to fire...if you shoot one lens length over and over and over, after a while, you know exactly what that lens will do.
It's like eating with a fork...you can put bites of food into your mouth every single time, without stabbing yourself in the eye or the cheek.