DNB
Your car is doing the E X A C T same thing that my 98 coupe did!!

Flustrating, isnt it!!
Here is what I did to solve that issue!
You have a bad ground or grounds. There are several grounds that you need to service but i will start with the ones that cause this problem.
Here is a post that i started a while back. It has helped numerous forum members solve lots of electrical issues.
http://forums.corvetteforum.com/show...ion&forum_id=1
The three that you need to work on are G-101, G-102 and G-401.
The older the car, the more likely that chassis ground issues will pop up.
PM me if you have any questions.
Bill Curlee