Actually I tend to do the same thing. My natural tendency is to shift to my right leg.
What helped was changing my squat technique. I used to the very quads dominant and would squat straight down, breaking at the knees first. My left hip flexor often felt a twinge and I shifted to the right side instinctively.
I widened my stance a bit and focused on rooting/digging/screwing my feet into the floor (trying to rotate them out) and initiated the eccentric by breaking at the hips first while continuing to screw my feet into the floor.
This morning I did 495lbs on the safety squat bar without any shift.
I also recommend strengthening your glutes.