iPhone 4 repeated force-close of apps

A few days ago when my girlfriend was using her iPhone 4, suddenly any app she opened would force close. Sometimes her music app would open on its own. No matter what she tried to do (reboot, backup and restore) the apps would continuely crash roughly 10-30 seconds after opening them. We went to the Apple store to have the Genius bar take a look at it. First they advised to restore and set up as a new device...we tried that, to no avail. Upon returning to the store a second time, they replaced her phone with a new device. When she got home and restored her backup the problems began occuring again.

Currently we are at the idea that her backup is corrupted and she should just start from scratch. The problem with that is I would still like to find out if it is an app that is causing the corrupted backup and if backups can be recovered. The most important thing is I want to find out if there is something that can be done that she doesn't loose anything/everything.