shimmy upon acceleration

I just bought a 92' Prelude SI. The motor was replaced 30K miles ago along with the clutch. The car itself has 203k miles on it.

When I accelerate in a straight line, the car shimmys/shutters about 80% of the time. It happens in lower gears and at slow to moderate speeds (it's fine on the highway). If I have the wheel turned one way or the other (like around a turn or bend in the road), there is no shimmy upon acceleration from a stop.

I tried to analyze the problem today and this is what I noticed. With the car stopped in neutral, the engine does not move very much at all. When I pulled up the e-brake and let the clutch out slightly, there was a good bit of movement in the top motor mount. I have never owned a Prelude before, so I don't know how much movement is too much.

I don't want to put any ideas into anyone's head. I don't know if I have a bad motor mount. Maybe it's just a symptom or result of a bigger issue. Maybe the motor mount is fine and there is something loose in the front end. I just don't know. If it was a bad motor mount, I'm not sure why turning the wheel would make a difference. Suggestions and feedback, please!

Thanks!