this is the watch of Adam
..is not a natural English sentence.
If you are going to put the words in that order, I would say "This watch is Adam's".
I know one Adam from Poland. He's cool person, and because of him I'm here!
Good for you! :)
"This is Adam's handwatch"?
I don't believe that "handwatch" is commonly used in English. "wristwatch" is, though.
What is wrong with "This watch is Adam's"?
Mostly the fact that the subject of your sentence is completely different ("this watch" rather than "this/that/it").