Well thanks to all. You've given me a lot to think about and explore as I track down a solution.
FWIW, moving the axle rearward in the dropouts -- and I moved it all the way back -- didn't resolve things. I was hoping it would, as the weight was effectively moved slightly forward. No dice.
In the meantime I suppose I'll keep my hands on the bars.
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A race bike in any era is a highly personal choice that at its "best" balances the requirements of fit, weight, handling, durability and cost tempered by the willingness to toss it and oneself down the pavement at considerable speed. ~Bandera