It sounds like frame size and saddle height/angle have been sorted.
How about forward reach? The length of the stem and the elevation of the handlebars in relation to the saddle will make a tremendous difference on the positon, and will influence how much weight is on your hands.

The grips will help, for sure, but if you are leaning too far forward your hands will still be bearing too much of the weight balance.

For starters, I'd recommend getting the bars at or higher than seat level, and possibly a shorter stem (or one with a more dramatic rise and not as much reach).