Also, check that any "play" is actually in the headset by placing a finger around the top race on the headset and rocking the bike back and forward while holding the front brake. Sometimes play in the brake can make you feel a slight knocking or bumping.
If it's loose then binding and there's no sweet spot, then it's either the headtube is out of square (unlikely) or a cup isn't fully seated into the head tube (unlikely) or like Splatter Paint said, something is either upside down or there is a seal jamming.
If it's the original headset then it is threaded. You need two spanners, one to hold the top race in place while you somehow stop the forks from turning and then tighten down the locking with the other. Then if you're anything like me, you still get it wrong, repeat, swear, curse, repeat and finally get it right.
Good luck with it!