yeah, that was just about exactly how my bird felt before I started to upgrade and improve the suspension.
here is a quick list of what I did or would recommend:
Get some good lowering springs all the way around, I like the hotchkis TVS kit and bilstein shocks from hotchkis also
A larger front bar (and maybe a rear bar too) included in the TVS kit
subframe connectors and solid body bushings
"3rd gen" steering upgrade- will take the loosey goosey power steering and make it firmer and give it more road feel (this would be one of the options for a stiffer box David Pozzi was referring to)
new bushings on the control arms and replace worn steering linkage and rag joint
g-mod (use the search function here, it is well worth the time and it is free)
modern tires with grip
there are a ton of different ways of getting there but improving the handling is well worth it and made my car much more fun to drive
Agree 100% with all of this, as well as with David on the floor brace.
Putting the shakers on will probably kill that vibration.
It sounds like you have achieved the ride and handling of a new early Camaro. OK by '60's standards, nothing like cars today.
The thirdgen box will give you firmer steering with more feel, see the sticky at the top of the suspension forum.
The Guldstrand mod is free, and will vastly improve how the car feels cornering. It will turn in solidly and hold, and won't feel like the tail is wagging the dog thru the corner. See Davids site for details.
Note: This is one that REALLY REALLY works. You can feel the difference even with big old soft 15" tires on there.
Bigger sway bar will help even out the body rolls, addco is good and ius cheap, hotchkiss is good, but not cheap.
Rolling Stock. The ONLY way to truly achieve modern handling in a forty year old car is with modrern rubber. 16 or 17", 40, 45, or 50 series performance tires. These cars like a lil more rubber on the back than the front/