What happens when you use your fingertips on teh bars? Back gets awfully tired?
If you can, get someone to take some pictures of you riding on the bike and get close-ups of the hands so we can see what kind of grip you have on the bars. Gloves with padding will just alleviate the symptoms (too much pressure on the wrong spots), but does nothing to remove the problem of having your hands oriented in a painful position on the bars. I actually ride 70-miles+ without gloves on a bare bar with no tape or padding; although most of the time is spent on the hoods or drops.
Do you have any symptoms of RSI or carpal-tunnel syndrome? Could be a pre-existing condition and the bike-riding just exacerbates it..