It sounds like you have a cold, maybe the flu; most vaccine allergies have to do with itching at the site, I don't know any that have coughs at all. I'm glad the Advil helps some. I'm sorry you feel so lousy and for so long. But I bet you are halfway through; it should be gone in another week or two.

I think you have to have the travel shots; you might not be allowed back in the country without them and the new avian flu kills something like one in 4 or 5 people who get infected; I wouldn't play with that.

Health Information for Travelers to Taiwan - Travelers' Health - CDC