Also, it would help you a lot to get your car professional cleaned. That way you don't need to touch the fibreglass anymore (no more allergy/asthma symptoms) and the car would be clean when you get it back.
Or maybe at the least, pay a friend to clean it for you to limit your contact with it.
Fiberglas will make the throat itch and be dry. Many people break out with a rash from fiberglas. I suggest you have the headliner replaced as soon as possible or face continuing problems.
For the itching on your skin, you need to take a shower in this manner. Start with a comfortable water temperature, and turn it colder little by little. Let it get as cold as you can tolerate it. Use soap and cloth sponge or another bath aid. to completely wash your body.
The glass will exit your skin and you'll feel better soon. Its the cold water that opens the pores and lets the glass particles go free.