Since you take UP TO two actions, and searching is not an exposure action. Couldn't everyone just search and not take a second action until everybody knew both clues?
Is there a rule that prevents this that I missed?
Sure, you could do this! But if you do, you probably won't make it past the first Act! It depends on the number of players, I think. With more players, this strategy could work - for a while. But then making it through the 2nd Act would be very difficult.