I can't tell you why photo stream photos are deleting from your old phone if it is still connected to photo stream. This should only happen if another device is sharing the same iCloud account and the photos are being deleted from the other device.
The best way to preserve the remaining photos is to save them to the camera roll now (by opening the my photo stream album, tap Edit, tap all the photos, tap Share, tap Save to Camera Roll), then import them to your computer (http://support.apple.com/kb/HT4083).
Backing up to iTunes would not have been any different than backing up to iCloud. In both cases, only photos stored in the camera roll album are included in the backup, not photos in the photo stream album. The only way to be sure they are saved is to import them from the camera roll to your computer via usb as outlined in the linked article.