Shaky hands are definitely linked to alcohol. Mine were always a little shaky, even before I drank, but they got really bad with alcohol. That went away when I quit too (mostly... I still wouldn't try to be a surgeon!).
Don't worry about the drinking dreams. I had a lot of them at first too. I actually took them as a good sign... it meant my brain was really focused on how much I didn't want to drink. I still get them maybe once every two weeks or so (I'm 10 months sober). when I wake up I'm just that much more grateful.
Congratulations on your sober time!!