Why does my car shake or vibrate when it changes gears? What do you recommend I do?

I have a 1994 Mercury Grand Marquis (same as Ford Crown Victoria) with an automatic transmission. When it shifts either from 3rd to 4th, or 4th to Overdrive (I'm not sure) the car vibrates hard for about a second. It feels exactly like I'm running over those white speed strips/bumps that are sometimes on the freeway. I am scared that I may be close to ruining the transmission, but I am afraid to get the transmission flushed, as the car has between 110,000 and 150,000 miles on it. I feel like flushing may fix the problem, but I hear that it may destroy the .