The point of hand positions is that when you start riding longer distances you'll need to change positions to stay comfortable. Drops give you at least 3 positions.

It's possible your bike just doesn't fit you right or it's possible that you just need to ride more and get your body conditioned to riding the bike. Did you buy the bike form a shop and get fitted by them? If you want to be more upright you there are some changes like flipping stem to face up (if not already) and/or putting more spacers beneath the stem to raise the bars a bit. Drop bars also come in different widths. Most bikes are sold with something neutral like 42 cm that will work for the greatest majority of people but if you have very broad shoulders you might need wider set of bars. The hoods are where most people spend most of their time. That gives you a neutral wrist position that will be comfortable for long ride. I like to ride on the flats of the bars when I'm doing long climbs or long flat straights (basically times when I don't have to worry about turning or braking)