Your weight should not be resting on your hands. The way I check this is:
1) ride with your hands in the drops (lower part of the bar)
2) move both hands at once up to the tops (flat, upper part of the bar)
If you have to "push" yourself up to that position instead of just using your core to "sit" up into it, then you've got too much weight on your hands.
If you're not familiar with other aspects of bike fitting (saddle height, fore/aft saddle adjustment, proper knee bend, etc.) ask how much your local shop charges for a basic bike fit. Shouldn't be more than $25 and should take less than 15 minutes. For a 26mi daily trip, it's well worth the money.
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"I feel like my world was classier before I found cyclocross."
- Mandi M.