I HAD one like that, I got horrible headaches, too. I set it in the sun for some days, I washed it with baking soda, I washed it with vinegar, I left it in the sun on a flat surface for several days in a row, switching sides. Never got rid of the smell so I traded it to someone for a smaller one, just to get it out of the house!
You may find a better answer here somewhere, these folks are amazing! I got rid of mine before I joined here so I didn't have the experts to help. whahhhh
Good luck,
Sharon