Before I get started too deeply in this, I wish it to be known that my wife and I dated. We, like you, were raised in a culture that taught dating. We never gave it a second thought. As I've gotten older and now have two grown daughters of my own, I have begun to re-examine some things.
First of all, in the history of the western world, dating is a reletively new concept. In fact, it really only appeared as a strong cultural movement in the early 1900's- not truly becoming a cultural mainstay until the "roaring 20's". Even then, it was mainly in the big cities that tend to be more liberal than the more rural areas. By the 50's it had taken over in even those areas.
Before this, Courting had been the cultural norm. Courting allowed a perspective "couple" to get to know each other, within limits. It allowed them to not only say, "Hey, you're cute. I want to know you better". But, it also allowed a safe way for them to get to know the person behind the looks and see if the person could possible be the person God intended for them to marry.
Why do I say safe? Let me explain: In courting, there are "fences" involved to help keep the couple pure before God. Not so in dating. In courting, the couple can be alone: on the phone, online, on the front porch, at a restaurant, at church- but with none of the "chances" taken in dating. A courting couple can sit on the front porch, or in the back yard, or in the living room. However, they are never completely alone. Always, a parent, or an elder, or another responsible person is nearby. Not near enough to overhear the couples conversations, mind you. But, near enough to cause a hindrance to the raging hormones that so often take over in a "dating" relationship. There is always someone else there to help keep you accountable.
Dating, tends to be more casual. They go out to "have a good time". They like to "have someone", but still have their "freedom". It is not unsusual at all for people to date many different people at the same time, just for fun. No commitment involved. Or, commitment as long as the other person does what they want. There is WAY too much "freedom" (or so the world calls it. God calls it bondage) involved in dating. The only thing holding back a dating couple from sin, is their own tenuous hold on themselves. Most of the time, in the heat of the moment, they lose their grip on that hold.
Courting says, "Lord, they're cute. If you'll help us and give your OK, I'd like to see if this is the one for me." This isn't trying each other out. And it doesn't include sex. Sex comes AFTER marriage, not before. Living under the same roof as a couple comes AFTER marriage, not before. The conversations in courting should include how you think, how you act, how you believe. Things in common. Things different. Are these differences complimentary? Or are they deal breakers? How do you feel about children, marriage structure, money, homes, where you'd like to live, etc. Each others personal histories (good and bad).
Dating is "rolling the dice and hoping everything works out for the best".
Courting is planned, deliberate, PRAYERFUL, and thoughtful.
Irregardless of our "grown up rights", Courting is BY FAR the better way.