How terrible for you! I've never had that happen, but petting a dog with Frontline on it makes my hands turn red and swell up like balloons. In your case, it's also possible that it's one of the other ingredient in the starter. Is there any possibility that it could be the litter or chicks? Or did it start as soon as you touched the starter? If it really is the medication in the starter, you could easily go with unmedicated. It would be good if you could handle some, first though. Just to make sure it's not something else in the feed.
I hope you figure it out. Do you have any other allergies?