I am honestly more of the mindset that you are not quad dominant and that weak quads lead to knees drifting in which is contrary to some of what you read on the internet but I have hands-on experience addressing this issue by adding in higher volume front squats as an accesory, and was told this by someone who is a much more esteemed and talented coach than myself.
My thoughts are that your total leg strength needs to come up. Just keep training and add in some movements (personally I like front squats as an accesory movement 6-8 reps after the 1st movement) and see what happens.
Also, as I am sure you are aware, you increase your range of motion significantly and basically force yourself to stand up with the weight as opposed to driving the hips forward when this happens making it very difficult to transition out of the hole which is where you got smashed on your 3rd.
On your work sets of back squats I would really focus on leading out of the hole with your belly. Try really hard to place emphasis on what I call ‘closing the hips’ once you come out of the hole. It’s like a pull where you want to emphasize the forward movement of the hips and end with your ass squeezed hard…2 people who come to mind that do this very well are Tylutki and Ribic so check some of their video and I think you will see what I mean.
I also think you need to move in your stance a little. Like an inch a side.