There are programs you can use to back up and restore select items from an iDevice (iExplorer is one example) however most cost money.
I face the same thing as you every year - I always want to start fresh but don't want to lose app data from apps that don't save their data to iCloud. As a result, I always end up restoring from a backup, year after year. Pretty sure there are legacy files somewhere within the file system from my iPhone 3GS since I've been restoring since then.