If mount says you have sda1 mounted at /, then you are running from sda1. Of course you need to change fstab to your new partition.
I think you need an archieve flag with cp. I haven't transfered a system with cp, but I have many times used 'rsync -aXv'. I have written about it in another thread
Mount both your new root and new home, and then run rsync. There is no problem at all that they are not next to each other or to your old root.
mount /dev/sdXY /mnt/new
mount /dev/sdXY /mnt/new/home