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Here is how I handle a stop. I am right handed.
I keep my wallet in my left front pocket. I often have a firearm in the door side pocket. My carry gun is IWB and insurance papers tucked inside the visor.
When I'm first lit up. I take my wallet out of my pocket with my left hand and set it on the dash. Not a lot of shifting or moving around involved. I wait until he approaches and I can tell which side he is coming to and roll down the appropriate window. When he is at the window I reach for the wallet, and hand both the license and the LTC to him. I then pull the insurance paper to hand it to them. Most of the time they don't seem interested in the insurance papers. They usually ask if I'm carrying and where it is. I think that these days the cops aren't that concerned about CHL holders. I've never been disarmed, and one officer told me that he sees CHL holders as a backup and actually thanked me for me for carrying. I keep my hands high and make slow and deliberate movements.