I have the same problem, but right hand. A few things seem to work wonders: 1) I bought the thickest padded gloves I could find (of course, they cost about double thinner ones) 2) Then when I ride I shift several times drops, hoods, up, down and 3) I remove my right hand if I have to stop for a light or traffic, or when I take a drink from the H2O...
I understand the addiction...had my road bike 9 days, have over 100 miles...LOVE IT... (my hand problem occurs MTB as well as road...)
Good luck...