Welcome to the forums. Congrats on the new bike and I'm glad you're alright.
It sounds like to me that the bike is leaning just a little bit on you and then it's too much to hold up at a dead stop. Don't sweat it, it happens - especially when you are just starting out on a heavy bike like the six. What year is it?
I had a 93 ZX6D for sometime. I had several years of riding under my belt by the time I bought it. I dropped mine too and at a near stand still. It's a heavy bike and you really have to concentrate on keeping it upright when stopping. Right now your mind is very busy with the controls of the bike when you are riding. You are concentrating less on keeping it upright and more say on getting it in the right gear or traffic around you.
This happens alot in the first couple of rides with some new owners... don't feel bad it's not uncommon. One of my past girlfriends used to drop my Ninja 250 (When I had it) everytime she came to a stop. Everytime. A little patience and persaverance (SP??) and she was fine.
I think what Pete said is a good idea, take your bike to a parking lot and get used to it's weight when coming to a stop. Go up a couple gears, then go down and stop. If you do it a few times, it will sit in the back of your mind when you have to worry about other things.
Don't get discouraged, you just need to get a little time under your belt. Accept that it may happen again and don't fear that possibility and you probably won't drop it again. I think every newbie I've spoken to has somehow dropped their bike in the first week or two. I don't think you need to change bikes. Just practice a little and try not to worry too much.
Plus, I live close to ya and I do fairing repair