I had a shallow hip hinge similar to the OPs that I only even realized when I went to my certification. A cue that helped me from my instructor was to "squat your hinge" more. While we are told the hinge is not a squat, my instructors realized that, in my case, that cue resulted in a shallow hinge.
This article by Rif also helped me a lot, as well as
this one by Brett Jones.
My instructor had me stand a couple of feet in front of a wall, facing away from the wall, and hinging until my rear touched the wall. I had to play with the distance until I could barely hinge back far enough to touch the wall. Then, I had to practice that deep hinge 100 + times per day until it became automatic. Then I built that hinge into the basic kettlebell movements, starting with the dead stop swing and then ending with the snatch.
I found that a deep hinge took a lot of strain off my arm during the snatch. I didn't have to pull the bell up with my arm anymore. Rather, the bell was being launched out AND UP due to the deep hinge. My arm only needed to tame the arc (agree with the towel drill as well to help there) and catch the bell at the float. Also, on the way down, a deep hinge allows you to "catch" the bell with more shock absorption, also taking strain off your arm.
Another cue that will help is to think of the explosion out of a deep hinge as a "jump that doesn't leave the ground." Focus all that stored energy into an explosion from your knees down through your midfoot. You will be amazed at how much lighter those bells will feel and how much stronger your are. And, hopefully, your elbows will thank you.
Best of luck.
John