Hello~It sounds to me like you are allergic to one of the chemical flavors in the Popsicle, I would ask your doctor and in the meantime, just avoid them or try a different type of frozen treat and see if you have the same reaction.
This could be an allergy to a dye or flavoring in them.
Do you get this reaction to other cold foods, or other types of popsicles? If it's only those, then it's probably a reaction to something in them, and not just because they'd been in the freezer.