"I got on a few with drops, and even though the frames were my size, I felt like I was leaning way out, and the brakes seemed far away. Would I get used to this? Is this a better way to ride?"
You would have to describe how you have your straight bar hybrid bicycle set up to sort of determine if the drop handlebar bike was properly set up for you. I know several people with straight bar hybrid bicycles with adjustable stems turned up so the bar is almost directly above the fork steerer. They sit bolt upright on the hybrid bicycle. If they sat on a drop bar bike of the right frame size, they would be leaning over considerably more than they do now. They would need a very, very short, very, very tall stem to duplicate their hybrid bike position on a drop bar bike. Whether the leaning over position of the drop bar bike or the bolt upright position of the hybrid bike is correct is anyone's call.
A few years ago I talked to a man on CANDISC about straight and drop bars. He had purchased a bike and had the shop change the bars and shifters from drop to straight. He thought the drop bar bike position meant riding bent over with your nose on the ground and he thought it would be uncomfortable. I explained to him that where he had the grips on his straight bar were the same position I had my brake hoods on my drop bar bike. Roughly. But with the drop bar bike you also have a more upright position with your hands by the stem on the tops. And a slightly more bent over position with your hands in the hooks. More options for hand position with a drop bar bike. If you were to measure the position of the grips on a straight bar bike and the hoods on a drop bar bike owned by the same very experienced rider, you would find that they are in about the same comfortable position on both bikes. You can get the same comfortable, or uncomfortable, position with either handlebar.
A drop bar bike allows you to shift weight from the saddle to your hands. With a straight bar bike, most people are fairly upright. All of their weight is being supported by the saddle. With a drop bar bike, properly fitted, you will support some weight with your hands/shoulders/arms when riding with your hands on the hoods or hooks. You don't want to support too much weight with your hands of course. But for longer distances, getting some of the weight off of the saddle can be important for many people.