Doubletree Hotel is part of the Hilton Group and there are a number in the Miami / Fort Lauderdale area if this is where you are staying. Nice hotels but I am not fond of their food options but that is me not the problem with the hotel.
One of my conacts says this might be the Doubletree hotel you may stay in http://doubletree1.hilton.com/en_US/dt/hotel/FLLTSDT-Gallery-ONE-A-Doubletree-Guest-Suites-Hotel-Florida/index.do.
Looks like you will be overnighting in Newark NJ. The normal carriers used for Newark are BA and Virgin but as this is the start of the busy time I would suggest, as you have a booking, you ask RCI the name and the "Locator Number" or Reference Number you will have for your flight and then go to the actual airline web site and check it out there. I would also recommend that you look at a site called trip advisor ( all one word) as they are the best to find reliable and up to date hotel information from past customers who have stayed at that hotel. The port Terminal at Port Liberty is not the finest in the world and I am sure that if you have a fly and cruise deal that you will get transfers to the ship which is much better than a diy try.. Hope that helps. ..... Neil.