The new button unlocks at level 7. Level 6 tells me you are stuck in the Crushing stage with her. Did you do the talk option with her and do the more than friends option? Then ask her to be your girlfriend from there?
I'm not sure why your achievements wouldn't be showing up in the contact app for her if you are seeing the badges for this... I just did a speed run of it and didn't have an issue.
Can you do the more than friends TALK tree, then after it is finished submit a bug report from the settings menu and email it to animaugame@gmail.com?