Creakyknees

So I picked up a frame off ebay. Lugged steel Columbus, De Bernardi brand imported by Yellow Jersey in Madison WI. Riding it, I noticed it "pulls" to the left when no-hands. As in, when no-hands, the vertical axis of the bike tilts a few degrees to the left while going straight. So I have to cant my hips left a bit to keep the bike straight.

This makes it somewhat difficult to control hands free but is not noticeable with hands on the bars.

My first thought was, "cables pulling on the bars". So I unhooked all the cables except front brake (which obviously wouldn't cause any pulling) and rode again. Same pull, same tilt.

So I put it on my trainer. Not as accurate as a frame table of course, but reasonably straight. Now it's obvious, when riding the bike looking down, and especially when out of the saddle, that the head tube plane is a few degrees out of what vs the seat tube plane. Such that the front tire virtual contact patch is a good several centimeters left of the bike's centerline.

So. Is cold-setting a reasonable solution? Is it feasible for a careful home mechanic who knows his way around bikes? Methods? No idea how to set this up even if it's a good idea.... Alternatives?

thanks
Creak.