For new grips, I'd seriously consider the Ergon grips (or other flattened grips). Perhaps the ones with small bar-end stubs that can be used for additional hand positions.

I ride mostly drop bar road bikes, but haven't been beyond experimenting a bit.

Your back and torso should take up a lot of the weight that would otherwise be directed at the hands. One thing that is sometimes discussed is the balance point. Putting the seat further back puts more weight on the seat, and putting it further forward puts more weight on the hands.

In a couple of cases, I've found that a shorter reach at the bars actually puts more weight on my hands. I'm still trying to understand that, but perhaps it has to do with natural angles between the pedals, torso, and arms. Bent elbows?