As I've mentioned in another post, you should be able to feel like you can easily bias your body weight fore or aft at any given moment of the ride. Wrist pain should primarily hint towards poor fit causing too much weight to hang over the front of the bike. However, a combination of the following factors are likely to be involved:
1. A combination of different stabilizer muscles in your body, including your core and lower back, failing to maintain good posture
2. A combination of incorrect stem length and saddle position attempting to compensate for an overly long top tube or reach, or an overly short stack.
3. if the pain is this significant, unless you simply rest too much weight on your wrists due to insufficient support from your core, usually this would simply point toward a poorly fitting frame, to which not even the shortest of stems could alleviate.
This means, that you are more than likely in need of a different fitting frame, or at the very least ditch the long stem and drop bars before attempting to address the issue of wrist pain from a secondary perspective, such as installing cushier grips, using wrist braces etc.
I would suggest focusing on full body toning and strength workouts (full core and spinal erector strengthening at the very minimum) so that you can engage more of your stabilizer muscles while centering your weight into the cranks, using these muscles to improve balance/stability while keeping weight off your saddle/hands, which should merely be contact points for the purpose of surveying road surfaces;specifically around corners, rather than simply resting the brunt of your weight into,