I seem to naturally skip the girl part.

Try this:

Talk about something that is happening at that current situation, then follow through. It's quite natural to do and you kinda lose that "tension" you get when planning what to say. For example, in your chem class the lecturer might fall down. You comment on that to the girl next to you and then maybe ask them if they're bored. Then where they're from. Then what they're doing after this "boring" lecture.

Follow through, know what I mean?

Take the current situation, comment on it, capture her attention, quick queries about her then ask if she's available after.

Neat and clean. No mess, no fuss. If she say no then move along.

Jam it back in, in the dark.