Have you tried rotating the bars up or down and/or changing the position of the shifters on the bars?

There's a lot of different drop bar shapes and sizes. I couldn't adjust the stock bars on one of my bikes in a way that allowed me to be comfortable both in the drops and on the hoods. I got some different bars and it made a world of difference. However, the problem may not be your bars at all. It could be some other fit issue. There shouldn't be a ton of weight on your hands.

Anyway, this link might be helpful: Road Bar Geometry