Pictures of the bike would be nice. One with wheel straight, one with wheel turned to the left or right.

At first, I thought you might just have a loose headset which would give you a 'thunk' or 'clunk' sound when braking, along with some movement. But I think TejanoTrackie has called it.

check this out: Bicycle Steering Geometry

Dave Moulton's Blog - Dave Moulton's Bike Blog - Trail, fork rake, and a little bit of*history

and lots more reading here: Spectrum Cycles | Geometry

Do you feel the handling of the bike is too twitchy at low speeds? Can you ride no hands? The frame should not be moving down but may feel like it is because the wheel wants to turn so easily.

Again, pictures would be a big help. Also identify the fork; see if you find any info on it. What size frame do you have? The headtube angle changes as the frame size changes.