Shots consistently to one side

Hi
I have been struggling with a problem of my whole group going to one side of the target,  it happens on the left side and the right side. Once it starts the whole group will follow.  I will line up what I expect to be a good shot and while squeezing the trigger at the last second my entire arm with elbow locked will drift to the right or left and the rest of the shots will follow suit.
I'm guessing this is a stance problem,  I have moved my rear foot to accommodate and it doesn't seem to have an effect . 
Could this be a grip problem? 
It seems like once it starts I really concentrate and say ok my last four shots went left try to keep this straight and the fifth will join them. If this is a stance problem no sweat I'll work it out but if it's something obvious that I'm not seeing please let me know.
Thank you, Tom

I stayed at a Motel 6 so I think your moving focus from dot to paper, or vice versa if using a dot.

What does the pistol rose tell you?

Download a pistol rose shooting guide off the web and see what that says.  There are a couple variations, for both right and left hand shooters.  Figure out the one that is applicable to you and use that one.  that is probably what you are doing. 

Regards, 
Cranky