Frodo wasn't going anywhere until the Orcs had passed by!
Movie Frodo didn't behave right either. If he really wanted to give everyone the slip he would have put on the Ring like he did in the book:
"I must go now or I shall never go. I shan't get a chance again. I hate leaving them, and like this without any explanation. But surely they will understand. Sam will. And what else can I do?"Slowly he drew out the Ring and put it on once more. He vanished and passed down the hill, less than a rustle of the wind.