You can also apply your gift cards to your reservation balance online. I go to disneyworld.com (the full website) and log into
MDE. Then go to "My Reservations and Tickets." Where it shows your remaining balance, click "Make a Payment." For the "Amount to Pay Today," enter the amount of your gift card. Then scroll down to "add a Disney Gift Card," click, and enter the gift card number. If you've entered everything correctly, it will then the display the amount of the card and allow you to apply the payment to your account, up to the remaining balance on the reservation.
I just applied a whole stack of cards to my account yesterday. Since I don't have a Target card, I buy GCs at Sam's Club at a 5% discount using a rewards credit card. The savings are worth it!
Whoops, I just realized that the discussion is about room only reservations, not packages. Not sure if the procedure is the same for those. FWIW, I have paid off room only in advance over the phone with no problems too. Good luck -- hope it works out for you!