If the only problem that you are seeing is that you want to move one component and not the others, but you still want them to share the same geometry, then you should not use "paste new", because that will create a whole new component for each instance.
What you may be seeing is that there are two modes in the Move command: Move Component and Move Body. By default, the Move commmand comes up in Move Body mode:

In this mode, you are redefining the position of the body within the component. So, every instance of the component will show the same results - this is considered a geometry change, and because all instances of a component shares the same geometry, all instances are affected.
If you change to Move Components mode:

Then, what you are doing is to change the position of just that instance.
Here is a quick screencast that shows the difference. In this video, there are 3 instances of a simple component, which has one body. First, I do Move Body, and every instance updates. Then, I do Move Component, and you can see that only one instance moves:
Hope this helps,
Jeff Strater (Fusion development)
Jeff Strater
Engineering Director