Ok, it has been touched on but I can not stress this enough. You need to activate your core muscles. Bike fit it very important as well as it will shift your weight in the appropriate way and relieve some of the stress from your hands. So here are my suggestions, coming from someone who raced road bikes for over 13 years as well as currently riding a fully rigid single speed mtb. I am also a personal trainer (part time)
1. Check out the Ergon Grips as mentioned or the ESI Chunky grips. The Ergon design helps to alleviate pressure from the hands and were designed with folks having carple tunnel in mind. The ESI Chunky grips are a slightly thicker version of their Race grip (I have a set of the race grips myself) and are a type of foam that is quite grippy and spongy feeling.
2. Adjust your seat slightly as mentioned above. You may want to start with trying to raise the tip of the seat up a tad or even possibly sliding the seat rearward slightly.
3. You need to do some core exercises to help strengthen the muscles in your lower back and abs. My suggestion would be start with some crunches and some stiff legged dead lifts. (you can look up both exercises on You Tube for examples of how to properly perform these-note that you do not have to do the stiff legged deads with a barbell in the beginning).
Exercises like this will really help to strengthen you lower back and core (mid section) which in turn will help you to hold yourself up more and rely less on leaning on your arms. Also remember to always keep a slight bend in your elbows which allows your arms to work like a shock of its own.
Lastly, consider a pair of bar ends for your bars. This will allow you different hand positions to use. Changing out your grip frequently will allow for a little more relaxation in your hands and help to alleviate some fatigue.