Since you have the computer that had your iTunes library backed up, I would try to get the library file and the music files from that backup and move them to one of your other computers. This will be a lot easier if all of your music, etc. was managed by iTunes in a consolidated library on your old computer, but it could be done even if this isn't the case. Then you can load your existing library (I believe you hold the option key (Mac) or shift key (PC) while opening iTunes to select an alternate library). If you're using a different username on the the new computer, you may need to edit the library file in a text editor so that iTunes can find the actual music files. You may need to authorize your iTunes account on the new computer as well. Once you've done all this and it looks like your library is functioning (i.e. music files have the correct metadata and play correctly without errors) you can backup your existing phone. Then you just sync the new one to this library. I've gone through a similar process when I've moved from one computer to another in the past with success. I even did it when I first moved from a PC to a mac, though that required some extensive editing in the iTunes library xml file to get it to work. The big advantage to this is it backs up everything: music, playcounts and other metadata, pictures, apps, settings, SMS, everything. Setting up your new phone will be easy and everything from your existing phone will be intact if you use this method.
Alternatively, there are several programs out there that will allow you to pull data from your phone. I don't know the details of these programs. I know they would allow you to recover your music, but I don't know if they would let you make a full backup to load onto another phone like the process above would get you. I don't think these programs tend to be free either.
