iTunes insists on erasing my iPhone
I just ran into the same problem. Here's the solution:

There are "iPhone Managers" that Apple does not promote using but it works 100%. These "iPhone Managers" (the names I'll give you in a second) allow you to transfer your CONTACTS, MUSIC, APPS, APP DATA, APP SETTINGS, PHOTOS &amp; VIDEOS and a LOT more to your Mac or Windows computer. 

After you transfer whatever it is you want to keep a backup of to your Mac in a nicely labeled folder on your desktop (just in case the iTunes Manual Backup doesn't keep some of your settings or pictures, contacts, etc.), you can then ERASE AND SYNC and transfer everything from the folder you made on your desktop with all of your files and transfer it back.

This is the only way I know and am doing it as we speak.

I tried posting links but since I'm new it won't let me. You can try these names:


CopyTrans
iFunBox for Mac
FileApp


I hope this helps.