I mean dating vs. betrothal, or maybe it could be considered a rather strict form of courting. I don't really call it courting, as to me that implies seeking to win the other person's heart, something that I think should be for the most part saved until one is inside a committed relationship. That looks a little strange in writing...I might not be explaining it the best.
To me, I want to guard my heart to the fullest extent possible, and that involves not getting romantically involved with a man on any level to the best of my ability. I have of course felt some attraction to a few people over time, which I think is normal, but instead of dwelling on it and letting it become significant in my life, I try to just go ahead with life and keep my focus on God, which is primarily where I feel my focus should be, particularly as an unmarried woman. (1 Cor 7:32-35, singlehood being a time when we can uniquely focus our abilities on serving the Master, of course this should always be our focus, but we may have special opportunities now that we wouldn't have when running a busy household

).
I of course don't believe we should be living in the shadows or should never talk to guys, but I don't think it is unreasonable to believe that our almighty Creator will ensure that we meet the right person at the right time, without our needing to search them out. I have never had a boyfriend or talked to guys on a level deeper than casual conversation, but the lack of an emotional connection with someone doesn't mean I am not paying attention and observing people to get an idea of who they are, how they act, their standards, if they stick to their convictions, etc. I think this kind of objectivity may be easier in the absence of an emotional attachment, and most of the people who I have felt attracted to it turns out are not people I would consider marrying.
This is actually a common stance among many of the people who are like-minded faith wise (we are sort of Hebrew Roots-ish you might say), and we know many people who have gone about marriage in this way. It is culturally very different perhaps, but it is beautiful, and biblical, I believe. It's almost as if many of us were raised in a micro-culture in this regard, the young men are very respectful and reserved in regard to seeking a girl's attention, and likewise the girls are mindful and reserved in that way as well. That doesn't make things dull or uncomfortable, it's just a pleasant, friendly environment.
Ideally, (in a broad general sense, every case will be different in some ways), the young man will at some point come to the place where he feels that God has brought the right woman into his life. He will seek Yah's guidance through much prayer, and hopefully seeking counsel from his parents and perhaps another godly older man. Then he would approach the girl's father (regardless of how he gets to this point, I think a man should always, if possible, go through the parents first), and naturally, after getting to know him some, the father would (hopefully

) agree that this is the Father's leading, taking into consideration first and foremost spiritual compatibility. Then, the father tells the girl and asks if she would like to go ahead with this (typically a very big yes

), and then the two begin to talk and seek to get to know each other in a reasoned way (don't get me wrong, I'm not saying that I don't think emotions will be a part of this, but still at this point trying to take a hard look as whether they are like-minded). Then... betrothal, which is making a commitment (sort of like engagement, I guess) wherein marriage is the definite end, unless some major, unforeseen thing comes up which would necessitate breaking it off.
Inside this relationship, it is a safe and secure environment to win each other's hearts, the "courting" which should continue up to and through marriage. In doing things this way, there is no giving away little pieces of your heart to various people before finding the "one," there is no need to worry about a boyfriend/girlfriend leaving for someone else, etc, etc. Of course, nothing's perfect, but as different as this may seem, I believe it is a healthy and biblical way to go about it.
I'm not sure if this makes sense, maybe it just sounds totally weird?

In any case, this is somewhat of a summary of my views on this, I'm patently bad at making short, concise explanations. I am not judgmental of anyone's views, this is just how we see things.