You may be (very) unlucky like me- as I seem to react to LOTS of unstarchy foods. I have been married to my iodine bottle since March, but my pain levels are still waver a lot (though significantly better, most of the time).
An elimination diet has helped me A TON.... this summer I fasted and then added one food at a time.... and found out (BAM!!!), that cabbage was a monster for me... the minute I eliminated it, my pain went from a daily 6-4, to a daily 2-5 (rarely above 4). If I eat romaine lettuce, trout and celery, I can get ALMOST pain free, but I can not survive on that (obviously).... so I am still in search of my pain free state .... but I thought my experience with the elimination diet might help you.
Also, beware of cucumbers.... I only eat seedless cucumbers (and even need to cut the centers of those out sometimes), since I discovered that the seeds themselves can be EXTREMELY starchy.... it does not show up with iodine, unless you split them open.... then they can be crazy black.
Good luck,
Stacey