A lot of good advice already given here. I can second the 'core strength'. The key to moving around drop bars is not to prop yourself up by your hands; your core strength (back and stomach muscles) should be doing this. Your hands are for shifting, braking and control. Make sure you are not locking your elbows, they should be kept a little bent. This will aid comfort of the hands as your bent elbows will act like shock absorbers. Once you are comfortable like this, you should be able to move into and out of the drops as you are not 'leaning' (collapsing) on the bars and pushing down on one side when you lift your hand from the other.
Tzvia- rollin' slow...
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