I just figured this out finally.
The only way you can add songs to a playlist using your phone and not having to log into iTunes is this.
Go to the very last song in your playlist alphabetically. Mine actually was a recording that had numbers. But if you have 3000 songs, go to song number 3000. Play it (you can scroll to the end of it). Make sure your "repeat" and "shuffle" are off at the bottom of the song screen. Basically, you need to not have a "now playing" link at the top right of your screen when you are making a playlist. If there is a song "playing," you can't do this on your phone. And the only way I saw that I could get rid of the "now playing" song is by doing this.