I am glad you finally found someone you think is game, but I think you're a little bit confused about the difference between "face-sitting" and "smothering." While the two terms are often used interchangeably, strictly speaking, face-sitting is when a person squats above your face and demands oral pleasure. Smothering also involves sitting on somebody's face, but the purpose is oxygen deprivation.
Suffocation for the sake of erotic arousal is a controversial subject. There is an element of danger anytime you restrict somebody's breathing. Fortunately, smothering is one of the mildest forms of what we in the kink business call "breath play." There are far more extreme types of the aforementioned fetish, such as strangulation, auto-asphyxiation, and full-body mummification where someone is enveloped from head to toe in plastic wrap with only a small opening to breathe. Leave those advanced breath games to more experienced players.
You also said you're unsure how to approach your girlfriend. There are two ways you can handle this matter: You can ask her or con her. If she says no to your request, don't frown, just trick her into it. But start easy. You want her to be relaxed. The best way to get a woman into smothering is by worshipping her body, especially her ass.
So next time you see your beloved chickadee naked, compliment her gorgeous bottom. Most women go gaga for praise. Call her a goddess and then ask if you can admire her hot ass. She won't be able to say no. And once you're on your knees, kissing and stroking her buttocks, stick your face in between her ass cheeks, so she'll get used to you being there.
Now that your girl is comfy, lie on the bed and suggest that she sit on your face. This is the best position for light breath play. For one, you can please your girlfriend while she sits on top of you. And two, you'll get the best of both worlds: safe suffocation and maybe even a hand job. But make sure your girlfriend positions herself so your nose presses against her anus and your mouth is covered by the area between the asshole and the vagina.
Remember: Any sort of breath play, even smothering, can be risky. You're a novice, so I suggest you do a lot of research before trying it or even make an appointment with a professional dominatrix. If, however, you can't afford a pro domme (most of us are quite expensive) and do it with your girlfriend, be very careful. If at any moment you feel lightheaded, tap your girlfriend to warn her and tell her to stop. Oh, and one more thing, do not -- and I repeat do not -- experiment with any sort of breath play on your own. I don't want you to end up like David Carradine.