Philbo,
There is something known as "cowl shake" which may or may not affect your Reatta - not all convertibles have it. Before you can diagnose your car, you need to freshen all your 30 year old suspension parts . . . struts, shocks, ball joints, etc. Tires are also extremely important to the ride of your Reatta, and a fresh alignment as well.
The Reatta was designed with an extremely comfortable yet sporty ride - it handles like a true GT designed with a touring suspension. Not everybody that buys a used Reatta plans to keep it long term, but the ones that do and invest in replacing tires and suspension parts are rewarded with a long lasting great riding vehicle!
Enjoy your Reatta Convertible!!