Sounds like you are either suffering from bump steer (change in toe angle when the suspension moves up and down) can be caused from lowering too far or putting different wheels on. Or if it is floaty at higher speeds and you need to turn it could be worn out shocks.

Bounce the front of the car and if it rises and settles quickly shocks are all good, if it continues to bounce they may be worn out. If its the original suspension in the coupe the strut tube is just filled with oil, So under a lot of up and down it can aerate the oil making your shocks useless.

It would be hard to tell without seeing and driving the car to exactly pin point the issue, but that's a start.

The bound bump is just a bit of rubber stopping the strut from completely bottoming out.