Re: Moving with your center
You need to learn to connect these parts of your body before you do so while moving with connection. To create the feeling of connection through the arms and across the back first have the feeling of your elbows being gently pulled throughout your movement. Some teachers talk about the elbows being "heavy" and the back rounded, whatever imagery works but you should feel the connection being made from the elbows across the back in all movement.
That will help with the one specific thing (connecting your upper body). Working on moving from your center can be done through Shiko or zhan zhuang (standing like a post) plus some form of slow movement like tai chi or aiki taiso....focusing on maintaing that feeling you get from the standing. I'd recommend standing since it has more fringe benefits than Shiko ( for me).
...and if you practice 3 or more hours a day you will start to pick it up over the next few months.
Since I'm wandering a bit in giving some advice...You might also try doing funokogi undo (rowing exercise) daily, with the same feelings. It should look and feel like the fajin exercises the Chinese guys do.