When I want stability and control at high speed, such as when I'm flying down a mountain on my road bike, drops are THE place to be. Drop are also great for fighting the wind.

Not even the pros ride in the drops all the time, most of the time they are on the hoods, as though they were riding bullhorns.

Track riders are always in the drops but that's a whole different scene.

Most likely, you aren't used to the position or your core muscles aren't used to it.

Fitted correctly on a bike, you should be able to sit comfortably with your hands on the hoods or just before the hoods on a drop bar. Then, take your hands away but don't move your torso. If you can hold that position comfortably, your fit is set correctly. If you feel like you'll fall forward, you need to adjust either the stem length or height/angle, saddle fore/aft, and also tone up those core muscles, both stomach and back.