What you omitted is if your new employer has filed and gotten approval for your H-1B through him (H1B is specific to an employer and not transferable, though you can skip some of the duplicate steps in the process if you already have one). If not, the new employer can not do anything about green cards.
You should not create multiple threads on the same general subject. Your case is complex enough that you should be talking to an immigration attorney. If you go at this willy nilly you can find yourself out of status in a hurry and back at the end of a long queue.

