Ummm, by this time Frodo realized that wearing the ring drew the attention of Sauron-and others, like black riders; that 'wearing' it was in fact, like wearing a 'magnet', making him visible to forces worse than those he could see.
Don't know, but he had been told not to use it, when he did...he had been wounded...
Yeah, putting on the ring would draw the Eye towards him...it already had just a few minutes before. But at that point, the Black Riders were nowhere near and Frodo (at least in the book) decided it was worth the risk to protect his friends. And it would only be until he could slip away.