You don't necessarily HAVE to shuffle your feet out too much on the catch. If it's not comfortable for you then that's okay; it may allow you to get a little deeper in the receiving position though. If it's more comfortable for you to not move your feet then maybe try starting with your feet a little further apart. That's just a suggestion and something you can play around with.
I wouldn't say that it's upper-back strength on why you chest is dropping slightly on the way up. Try to focus on squatting with your hips straight down and then keeping them in, underneath your body as you ascend. You can see that when you Back Squat it's the same issue. You're bringing the hips back a little bit as you descend and that makes it much more difficult to stand with your hips in.
If you detail the position of the hips in relation to your torso, you can help that Clean in the same way.