You cant just change the folder names of Steam.
The shortcuts and filepath associations get coded based on where it is. Essentially you need to let steam know every time you change.
I think your best bet might be reinstall steam in that new directory, and then drag the game folders (common, drisks, etc) into the new directory from the fresh install.
Theres a way to do it with clientregistry.blob and clearing the whole steam folder but I've lost save data before doing this so I'd play it safe and keep all your stuff backed up .