New shoes will take some getting used to. If you look at the bottom of rental shoes they are (almost always) entirely leather except for the heel. Most bowling shoes the slide portion is only on the sole of the foot ending just past the balls of your feet.

So if you learned to slide more whole-footed with rental shoes then you will get that catching feeling with your new shoes. You'll probably need to learn to slide more on your toes and ball of your foot.

An option that may help is a shoe slide. I've been bowling with a shoe slide because my shoes are too tacky. The shoe slide tends to cover more of the shoe and might help you not to catch. Though it may not help in the long run to become reliant on a shoe slide. https://www.amazon.com/bowlingball-com-Shoe-Slider/dp/B004KQ75ZS/ref=sr_1_1?s=sporting-goods&ie=UTF8&qid=1510862214&sr=1-1&keywords=shoe+slide