The palms of my hands really hurt
You are correct that a leant-forward riding position will naturally tend to make a rider support some of their weight with their arms and hands. The more rotated forward the rider is, the more weigh the arms tend to have to support - which is one reason tri-bars have forearm rests.

Good core strength is what allows a rider to take weight off the arms. Strong lower back and abdominal muscles support the torso, taking weight off of the arms. You can try doing some core strength building exercises (which will likely make you a stronger rider and help prevent injury as well). 

You should look at your fit on the bike, you may in fact have the bars too far forward and down. Raising and pulling them back may help while you build core strength.