The first Cobra I built was a Classics Roadsters - nice car. You're right, it rides like a buckboard. The problem is that the rear of the car is light weight, and the axle is heavy - comparativly speaking. The similar weights of the sprung and unsprung weight makes it a little more difficult to control the suspension.
The first thing to do is realize that you're going to have to spend some money to get what you want. So, don't be shy with the check book.
Dump the shocks and springs you have now, and replace them with a good set of coil overs. Spend some time measuring your suspension, and then spend some time studying the QA-A web site. QA-1 makes a fine shock, but their springs are average. I prefer Eibach springs.