Dean,
I've been thinking about this bike.
First off, the more I think about the steering the more I like it and appreciate the design.
I noticed you built the frame out of round tubing, and this is probably part of the flex problem. As we all know, round tubing provides equal strength and stiffness in all directions, and in this application isn't what you want. You have basically built an I beam which is stiff and strong, but only in one direction.
You're frame looks like this:
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and the seat/head rest doesn't look to add any stiffness.
just an idea for a frame(not to scale by any means!):
/|
/ |
/ |
/ |
/_____|
Where the height of the vertical section is greatly exaggerated due to my ASCII art. I think if it puts your head at the same height or only slightly higher than it already is will add to the stiffness significantly (i.e. like maybe 6" at most) This would put your head at a more natural/comfortable angle as well.
Don't most of the 32 Ford frames use 2x3 rectangular stock? Seems like that would be overkill in your application, but since you can't make any structure that covers your body, might need it.
Just my $0.02, and hope you don't mind the unsolicited advice.