1988 Cadillac Deville Vibration "off the line"

For the past few days, when starting from a complete stop and easing on the accelerator, there is a strong vibration/shaking you can feel through the entire car. It's generally very brief. The car sits at an idle just fine. It also drives perfectly smooth at any speed once going. It also is perfectly smooth coming back to a stop. The hard vibration/shaking is just very brief (though fairly hard/strong) as you begin 'off the line.' It has only been doing it for the past 3 to 4 days. Yesterday, it happened maybe 4 times throughout the entire day. Today, it happened many times in only a ten minute drive to the store and back (so it's getting much worse very rapidly.)