Hi buddy,
Not a noob question at all, I've been using spotify premium for about 2 years now and even I'm confused by there new functionality. You see, the "Songs" and "My Music" section is a new addition.
But, how it's supposed to work for offline mode, or how it used to atleast, is that you add your songs to either existing playlists or new playlists. You then make these available for offline. Spotify will then start to sync these to your device at a quality you've selected in the settings.
If you added tracks to an existing playlist, or in your case "My Songs", that has already been made available for offline, the song will proceed to download to your device after you've added it, providing on a couple of things. 1) You have an internet connection and 2) your device isn't in offline mode. And presumably it wouldn't be the latter as you wouldn't be able to search for songs to add to a playlist in the first place.
After all this is done, you can turn on "offline" mode in the settings. What this ensures, is that you will only have access to the files you synced to your device. You can still play these out and about without opting to turn "offline" mode on, but whether or not it streams the track you downloaded or plays it from its offline cache I'm always a bit dubious about.
But that's it in a nutshell. I do exactly the same thing and listen to Spotify primarily in the car over Bluetooth using my iphone 5
🙂I hope this helps!