DOOM: The Boardgame FAQ, Berserk wrote:
Q: How long does the berserk drug last? Is it “the turn you pick it up and the following turn” or “the turn you pick it up and two more turns”?
A: The turn you pick it up and two more turns. I should also note that, due to a printing error, the berserk tokens aren’t faded on one side. I recommend either marking them yourself, or just taking two and discarding one instead of flipping it over. There are no scenarios in the booklet where you’ll be using too many berserks at once for this.
So you are playing correctly. =D