I solved problems like this by going to a longer stem and moving my fore-aft seat position to get more weight on my butt. I try to avoid padded handlebars because sometimes that presses the padding up into your carpal tunnel. Bike gloves have a gap there, but it's not always effective
I'm guessing your saddle to bar drop isn't helping you much here, you're probably putting too much weight on your hands. I would try adjusting the seat first.