Your stance and the amount of bend in your hips and knees can have a dramatic effect on your accuracy when doing a "run and gun".
You need to find a place to practice running and dry firing for starters... some place where people don't think you are insane or dangerous.
Then you figure out what is comfortable and efficient for you. A little bend in your hips and knees, with your weight slightly forward will move you effectively and serve to absorb accuracy damaging impacts. Practice, practice, practice. A comfortable, stable gait is important.
Second phase: What I used to do (and this is me... maybe not for you) for dry practice was to install a bore sight laser in the barrel, and set up some targets... or have your shooting partner set up the targets and course of fire for you.
You run and gun... when you think you have a good sight picture, yell "bang" and your partner can look at where the laser "hits" on the target. Repeat, repeat, repeat. Partner critiques. Don't have a partner? That makes it a little harder 'cause you have to critique your own progress.
Concern yourself with being smooth and comfortable with your gait and gun control... not fast, just smooth and accurate.
I make it sound easier than it is... it takes a whole lot of practice, sweat and yelling "bang" to get good at it.