Ron, you are definitely right about needing some new shocks, given the bounce test results. I am a firm believer in doing one change at a time and testing for acceptable results. The thing to do is work toward a better handling car without turning it into the proverbial coal cart and ruin the everyday driving enjoyment.
Once your get the shock replacement done, I have a couple of additional suggestions.
I read somewhere, perhaps on this Forum, an interesting observation attributed to Colin Chapman, of Lotus fame. He said something to the effect that any suspension can be made to work reasonably/effectively well if you can just keep it from moving. Or sentiments to that effect. That is particularly relevant to our Alpines and Tigers and the A-arm configuration and how they move under suspension travel. The stock design does not give you the negative camber you want (at least quickly enough)to be generated when the car leans in a turn. You sort of end up on the edges of the front tires. I am guessing it is that phenomenon that is causing the car to not feel 100% stable. You are loosing grip at exactly the same time you are needing it. Stiffer springs and matching shocks (and bigger anti sway bar)will reduce the body roll in a turn, helping to keep more front rubber on the road. Its that Chapman thing…
Another thing you can do is dial in a little negative camber into your alignment, say 1/2 a degree, Keep positive toe in of 1/8 inch or less for acceptable straight ahead tracking Tire wear should not suffer materially, and the negative camber will give you a leg up on what the suspension will eventually start to generate.
Tires are also a major contributor. What size of wheel and tire are you running. A high sidewall tire on a 13 inch wheel is going to give you a lot more sidewall flex in a turn than a shorter sidewall tire. That is going to create a bit of squirming, sense of instability, relative to a shorter sidewall tire. But at some point the ride quality becomes too harsh.
Just some ideas, hope some of it is worthwhile!
Gene