Jane -
Our family has something called "essential tremor" - just a trembling that starts in our mid teens and gets worse over the years. Toward the end of his life, dad couldn't sign checks anymore or eat in public very well.
I think it contributes to me waking with the "hard shakes"... but only when I wake up suddenly - like with a ringing phone or someone unexpected in the room (husband used to work nights and I'd waken when he came home - it would take a few minutes for the shaking to subside). For me, the shaking can be so bad that it looks like I am having a little seizure or something... but the worst of it goes away in a minute or so.
My hands shake all the time, and if I am tired or over stressed, my head trembles a bit, too. My brother had it and my aunt and my dad... Doc told me that most folks just learn to live with it.
I agree with the others that a doc's advice is best, and I would be sure to mention the caffeine to the doc, too.... much as I love my coffee, it exacerbates some of this crap.
I wish you the best....