Speaking as an inexperienced beginner, too (so anything I say is probably *%^&*($" 😉 ) I found that I was constantly hesitating slightly when jumping between octaves, especially when descending, because I was desperately trying to make it smooth and not honk like a dying goose being eaten by a fox!
Although practise has obviously helped, I've found changing mouthpieces has helped massively as well. Every time I've bought a mpc with a larger opening than the previous one, the smother I've found the transition to be. And the smoother the transition, the less apprehensive I am of making the jump, Less apprehension means being more relaxed and more relaxed makes the transition smoother.
I'm not saying you need to buy another mpc, but rather the more comfortable you are with the sax, the easier it is. It just so happens that a mpc with a much bigger opening than I was using was something that really helped me feel more at one with my sax.
I've watched experienced players fly through the octaves so smoothly it makes your head spin. That's definitely practise and experience. 😉
One day...