I'm assuming your posture is part of it. JRoden is the dude for all things cycling, but in general - make sure that your elbows aren't locked straight (you're not straining against the handlebars with your arm) ... Keep a slight crook to the elbow so your arm is fairly relaxed, and don't have a death grip on the hoods or the bars. Shift your grip from time to time, and go no-hands ans shake them out if you need to (safely, of course). If you don't use cork tape, invest in that or something similar.
Unless there's something wrong with your physiology, it shouldn't be something you have to live with. I don't wear gloves - and never have problems on long rides. The more your ride, the more you'll feel confidence in balance - and keeping your "core" centered on the bike, so you don't need to feel so tied to controlling the bike with your arms.
Also - if your bike doesn't fit well (or you haven't been well-fitted to your bike) - you may definitely have issues.
See this thread for more on positioning on the bike. Many others here as well!
Sorry to repost these pictures again, but if you ride with your body weight resting on locked arms, you will get numb hands, crash more often, get neck pain and a sore croth from riding. Work and keeing the elbows bent and support your weight with your feet, so it moves you down the road, heres two examples of bad form and 2 OK form:
Thanks everyone! Purchased gloves, looked at jel wrap for beneath the tape but did not buy. I will also have the fit of the bike checked to see if the stem needs shortened. I was in the drop bars most of the ride. "Claw hand" is very descriptive. I had to left hand start the car all day. Not fun!