Originally Posted by SkinnyLegs

Don't agree at all with what has been said. Your problem has nothing at all to do with your flexibility or fit as suggested. Those bars are simply not meant to be angled up as high as you have them. They are designed so that the bottoms will be about parallel with the ground when positioned as intended. But you have them angled up way high so the tops are a bit more comfortable when riding in the hoods. This obviously makes the position in the drops awkward as you've experienced.

You want bars that are designed to have a flat top and smooth transition to STI hoods. These generally have a very tight radius bend. This will leave the angle in the drops in a much more manageable position and also reduce the reach to the levers from the drops. Off the top of my head the FSA Omega Compact, FSA Wing Pro Compact, Deda Supernatural, 3T Ergosum, 3T Ergonova, and Easton EC90 SLX3 all have the shape you are looking for.