Hip drive won’t become a problem until you get up around your max. Depth is good, tiny bit of ‘wink’ at the bottom but nothing to worry about.
Ditch the squishy shoes. Get some cheap canvas flat-soled shoes like Chuck Taylors. Put some 10lb plates under your heels if you need to. If you can afford to, grab some weightlifting shoes (for the elevated heel).
But your biggest problem is that your upper back is too relaxed. Grab the bar with as narrow of a grip as you can while still being able to get it in the groove, and keep your back tensed and your elbows down.