I work in the Credit Card industry, and I used to work at one of the three credit bureaus, although I don't know all the ins and outs of the FICO score. Very few people do know the details, it is a closely guarded secret. If you call any of the three bureaus, they won't tell you how things will impact your score, for two reasons. First, FICO doesn't like information like this made public, since it makes it easier to game the system. Second, no one you speak to at the bureaus will know anything useful.
Having said that, closing a card won't impact your score much, with one exception that I will elaborate on below. If the issuer closes your card, it will usually ding you quite a bit, since they usually close your card because you are a bad customer. If you open a card, your credit will be hurt somewhat less, but still more than if you close a card, since opening a card is a possible sign of financial distress. The only time that closing a card will really hurt you is if you carry a high balance on all your other cards. Closing a card will then increase your utilization (balance divided by credit limit). If your utilization gets high, your score will suffer greatly. If your other cards carry no balances, I would be very surprised if your score dropped by 5 points, if at all. Any drop should resolve itself within a few months.
It may be worth it to ask for the fee to be waived, but issuers are more reluctant to do that these days (we're feeling the financial pinch), and it would only be a one year fix. They aren't going to waive it completely for you. If you don't use the card, they probably won't waive the fee, as they don't have much incentive to keep you around.
As far as the new card/transfer card thing, it depends on the issuer and the product. There are some cards that we can just transfer to a different product without closing the card, and some that we can't. Since this is an airline card, though, I would guess that they will have to close it.
I would just close it, unless it is a US Airways, Alaska Airlines, Hawaiian Airlines, or Lufthansa card. In that case, I would encourage you to open up 3 or 4 more of them and carry very large balances. It would really help me out.
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