Personally I found that having my saddle too far forward was causing me to put too much weight on my hands. In general I have heard that you want to divide your weight evenly between your hands, saddle, and the pedals.

Specialized BG gloves are good. I made the mistake of getting my first pair too loose because I wanted to be able to take them off easily. Bad idea. They end up slipping all over the place and the pads don't contact the tape as they should.

Also Specialized Phat Bar tape is cushy and helped me aleviate shooting pains that I used to get in the hands.

I independently discovered the advice given by San Rensho and it's good. Reposturing this way may help you.

Overall, more training is only going to help you but definitely don't try to ride through numb hands. Your intuition to take your hands off the bars is a good one.