My Samsung Galaxy S III running Android 4.1 looks like it's going to die and I was wondering what's the quickest method to backup my Favourites from Chrome and stock so that I can copy them to a new device? Note: I don't have PC access and I'd like to keep the folders.
For Chrome bookmarks the easiest way is to enable Chrome bookmark syncing via your Google account. If this hasn't been automatically done, you can find the option to log in to your google account from Chrome in Menu -> Settings -> Sign in to Chrome. After this, your bookmarks will be synced to the cloud and you can simply repeat the process on the new device. Note that this also works for the desktop and iOS versions, if you'll ever need them.
You could use a dedicated backup app such as Backup SMS, Contacts & Apps:
This is a Backup Manager / backup Assistant which lets you maintain backups of various things such as Contacts, SMS, Calendars, Bookmarks, Apps & Call logs. All of this can be kept safely in your email ID & on your external memory. Also helps transfer data from one phone to another.
(emphasis mine). The quote is from the app description, and as my emphasis shows, this should be perfectly fitting your needs.