Bicycle trying to steer itself - possibly a bike ghost? Are those real?

I had a bike once that had some weirdness in the front-end. Turns out, the fork had cracked where the forks meet the headset. So, in addition the above, I'd recommend taking a good look at the welds and such and see if there are failures anywhere. You might need a magnifying glass - in my case, the cracks were barely visible when the bike was unloaded. Point is - that's an old bike, and minute stress fractures accumulate.

Possible problems, not in order: 1) As loriginedumonde says, a headset does need more than mere adjusting, it needs occasional servicing. Possibly the bearings are shot, or dirty or both. 2) The same applies to the bearings of the wheel itself. If those are too loose, the ensuing wobble might show itself in funny steering behavior. 3) As Betty's Table says, perhaps the wheels need to be trued. However the effect of non-true wheels usually is more rhythmic, not intermittent. My best guess is in fact that you're looking at a combination of mild cases of all three of these problems - statistically, this is also the most likely. If you have a good bike maintenance manual, you'll find ways to assess all three; it's a rather easy thing to do. And no, a good bike shop won't think you're crazy. They would diagnose and fix.

Duh, should have looked at the picture. No brake cables to speak of. Sorry 'bout that.

First, take it to the shop and ask them. Most likely, they'll rebuilt/refurbish/overhaul the headset. However, there's also the possibility that your frame or fork is bent as that can lead to some weird steering issues. Any decent mechanic should be able to check that out as well.

Make sure you check the REAR wheel as well. Subtle steering problems are sometimes related to the rear.

I had almost exactly this problem and it was the headset subtly binding on when I turned the handlebars.

1) regarding headset bearings, with this old of a bike you may need races also. A hard impact can dent the race as well as flat spot the bearing balls. 2) Are the new tires different in diameter? This can alter your http://www.erikv.com/blog/index.cfm/2007/12/28/Forks-and-front-end-geometry and change the handling.

If it stays straight when you pick up the bike, I'd say your headset is probably too tight. You want it just loose enough that it easily falls one way or the other.

For certain, take it to a shop and have everything checked out if anything, for safety's sake. If everything is fine, and you are still experiencing the issue, I once had a set of tires that wanted to track/follow every little groove in the pavement or path. Even the rear wheel would squirm out in order to follow nearly parallel almost insignificant cracks and grooves. Replacing the tires removed the behavior completely.