This sounds very much like a scam to me. Too good to be true? Listen to that little voice in your head! IMHO, if you had actually won a bona fide trip, you should not have to pay air fare or fees. The moment someone I don't know mentions to me that I HAVE to pay anything by credit card over the phone ASAP, I hang up. Please don't get sucked into this, at least not without making a thorough check (and it sounds like they're not giving you ample time to do a thorough check.) We can't be 100% sure it's a scam, but something just doesn't sound right here.
_________________________
When you don't like a thing, change it.
If you can't change it, change the way you think about it.

(Maya Angelou)