I'm planning on replacing the Bilistien shocks on my 89 CONVERTIBLE(different p/n?).

I replaced the entire spring and shock package with Vette Brakes sport set-up which came with stiffer springs and shocks. I took the rear shocks right back off because they were just too stiff even though I kept both front and rear springs. Front shocks were a little stiffer but not like the rears. It would bounce the rear-end right off the ground going over rail road tracks.  I eventually traded all that stuff away when I went to coil over double adjustables. You might want to talk to Bilstein about ones that are valved like the OEMs.

just a quick question. are you over inflating the tires? the place where i get my tires suggest to follow the inflation rate that is on the sticker on the vehicle and not what is on the tire which can be a lot different. just seems that that is a lot of pressure to drop.

  I do not want to sound like I'm bragging on my self but I am fairly well knowledgeable on suspensions and tire pressures and from the schools I've been to Eddie is correct with his information on the tire pressures. Normally the sticker on the door is the correct pressures to keep in the tires although one 'can' vary tire pressure 4# to 6# to get the best wear out of them. One wants all the tires to wear evenly across the tread surface.  If one changes more than that you can expect the opposite happening.

  All so, ''shaking'' or vibration is caused most of the time by a tire imbalance, not the tire pressures. Tire pressure may let you aware of the imbalance due to being harder with less give in the side wall. If tire balance is not out, which it only take 1/4 of an ounce. That 1/4 of an ounce turns into 5#'s of thrust at 62 mph. It doesn't take much to be out of balance. The other most commonly cause of tire imbalance is caused by a loose part or even a bent wheel.  

My '84 has the Z51(STIFF!) suspension.  When I got it the shocks were completely shot and it rode like a log wagon.  I put KYB's on it and the ride really improved.  It's still stiff, but now I don't lose teeth when going over tracks and potholes.