The backup can be done via iTunes as well and you then have a local backup. Also you can sync/backup your pics to photo stream or sync to the local PC and get them back. You dont really need to increase iCloud storage for photos if you save them to local PC or photostream.
I concur that you need to setup as new and not restore the backup at least to start. You can certainly restore it and restore your backup and see it it helps. It could very well be fine but as others have mentioned the backup may restore the issue. Another option is restore via iTunes, restore your backup, then go to general>reset>reset all settings (NOT erase all data). Select setup as new when the phone reboots.This will not delete your apps and data that you just restored but you have to manually reset any particular settings and wallpaper etc. If that does not help then you can always restore, setup as new and sync and manually install apps again as worst case. You have options. I would try the latter first since it will restore all your stuff like it was and if the problem is gone you win, if not you can still restore again and go through the manual stuff.
Also there is an option called DFU restore. you can search for the process but that is a low level restore and has a good success rate of solving corrupt software issues.