Hi, I've had my LS400 (91) for about 4 years now, and it always had a slight problem of shaking, but just recently the problem has become much more bothersome.
When I give the car gas, it feels fine. However, when you try to level off the speed, and remain at a constant speed, the car will shake forward to backward motion very hard.
My guess is that it's not receiving enough gas? Perhaps there is something blocking the gas line?
I do also notice (for the first time) a rattling sound when stopped. (Can't hear when driving).
I just replaced a bearing on the engine fan that had gone bad, other then that no work has been done on this car. It does have a bad idle control too though, but I don't know how that could affect the shaking at level speeds.