Inconsistent headset tightness and squeaking
STOP RIDING THAT BIKE

I'm concerned that your bike has internal problems and that something may fail rapidly (ie, you're experiencing the warning now)

You need to inspect the front steerer tube and head tube looking for anything wrong.


Firstly, clean and inspect the frame area around the headtube.  You're looking for cracks or any damage.
Take photos of how its assembled before starting, as a visual guide.
Start by removing the stem, either quill stem if its an old style or the top cap and stem if its more modern.
Hang the bars out of the way so there's no pressure on the brake/gear cables.
Remove any spacers, or undo the lock nut and crown nut.
pull out the bearings.  They may be in a cage (easy) or there may be dozens of tiny loose ones.   Put them aside to wash in degreaser.
Drop the fork down and out the bottom.  Be aware there's more tiny ball bearings in the bottom, again may be caged or loose.  
Clean grease off, and inspect the


steerer tube - the "handle" of the fork.  It should be straight, with no cracks or deterioration.  You should have a good look under a bright light, and use a steel ruler to look for any bends
fork crown - the bit that joins the tines/legs/blades of the fork to the steerer tube.
Ball bearings (look for non-round ones, discoloured ones, or chipped ones)
Bearing races - the running surface that the balls roll on.  These should be clean, smooth and bright.  There should be no dents or dips or damage.  There will be four, two in the bike's frame, one on the bottom of the fork, and one on the top big nut.
Head tube - use a torch/flashlight, and look down inside the frame for anything wrong.



If everything seems fine, clean and grease and reassemble in the reverse order.  You might like to fit new ball bearings at this time, they're pretty cheap from the nearest LBS.  Getting the tension right on the bearings can be a bit fiddly, but it should be smooth with no play when fully assembled.

You might benefit from removing the front brake or its cable.  Removing the front wheel will make it easier too.

Once completed, it should be perfect steering, provided there's no damaged parts gone back in.  If you find damaged parts please update your question with clear photos.